L(s) = 1 | − 6·9-s + 4·11-s − 4·23-s + 8·29-s + 16·43-s − 28·53-s + 40·67-s + 8·71-s + 20·79-s + 23·81-s − 24·99-s + 76·107-s + 12·109-s − 32·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 66·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 2·9-s + 1.20·11-s − 0.834·23-s + 1.48·29-s + 2.43·43-s − 3.84·53-s + 4.88·67-s + 0.949·71-s + 2.25·79-s + 23/9·81-s − 2.41·99-s + 7.34·107-s + 1.14·109-s − 3.01·113-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 5.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.441883354\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.441883354\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2 p T^{2} + 13 T^{4} + 64 T^{6} + 316 T^{8} + 64 p^{2} T^{10} + 13 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 2 T + 6 T^{2} - 4 T^{3} + 147 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 + 66 T^{2} + 2108 T^{4} + 44094 T^{6} + 666886 T^{8} + 44094 p^{2} T^{10} + 2108 p^{4} T^{12} + 66 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 + 52 T^{2} + 1613 T^{4} + 36440 T^{6} + 690556 T^{8} + 36440 p^{2} T^{10} + 1613 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 - 8 T^{2} - 135 T^{4} + 5888 T^{6} + 34928 T^{8} + 5888 p^{2} T^{10} - 135 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 2 T + 45 T^{2} - 18 T^{3} + 1004 T^{4} - 18 p T^{5} + 45 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 4 T + 31 T^{2} - 20 T^{3} - 176 T^{4} - 20 p T^{5} + 31 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 130 T^{2} + 8772 T^{4} + 418766 T^{6} + 15006422 T^{8} + 418766 p^{2} T^{10} + 8772 p^{4} T^{12} + 130 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 43 T^{2} - 92 T^{3} + 932 T^{4} - 92 p T^{5} + 43 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 190 T^{2} + 17621 T^{4} + 1097480 T^{6} + 51314980 T^{8} + 1097480 p^{2} T^{10} + 17621 p^{4} T^{12} + 190 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 8 T + 91 T^{2} - 552 T^{3} + 4028 T^{4} - 552 p T^{5} + 91 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 28 T^{2} + 6184 T^{4} - 76596 T^{6} + 16621838 T^{8} - 76596 p^{2} T^{10} + 6184 p^{4} T^{12} - 28 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 14 T + 208 T^{2} + 1906 T^{3} + 16942 T^{4} + 1906 p T^{5} + 208 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 190 T^{2} + 17400 T^{4} + 1425002 T^{6} + 100823438 T^{8} + 1425002 p^{2} T^{10} + 17400 p^{4} T^{12} + 190 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 + 266 T^{2} + 36236 T^{4} + 3305206 T^{6} + 227629030 T^{8} + 3305206 p^{2} T^{10} + 36236 p^{4} T^{12} + 266 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 20 T + 346 T^{2} - 3768 T^{3} + 36179 T^{4} - 3768 p T^{5} + 346 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 4 T + 179 T^{2} - 408 T^{3} + 14996 T^{4} - 408 p T^{5} + 179 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 270 T^{2} + 33397 T^{4} + 3003592 T^{6} + 235852580 T^{8} + 3003592 p^{2} T^{10} + 33397 p^{4} T^{12} + 270 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 10 T + 137 T^{2} + 182 T^{3} + 1440 T^{4} + 182 p T^{5} + 137 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 486 T^{2} + 114541 T^{4} + 16838816 T^{6} + 1678490780 T^{8} + 16838816 p^{2} T^{10} + 114541 p^{4} T^{12} + 486 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 + 270 T^{2} + 45653 T^{4} + 5475384 T^{6} + 548443204 T^{8} + 5475384 p^{2} T^{10} + 45653 p^{4} T^{12} + 270 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 + 390 T^{2} + 66256 T^{4} + 7045522 T^{6} + 654890654 T^{8} + 7045522 p^{2} T^{10} + 66256 p^{4} T^{12} + 390 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.22642728640231108350096066403, −2.91224626360531064333961719534, −2.73848262351091451067122265452, −2.68129724530052194972945765940, −2.63699848720636296384487113238, −2.49674757074081814029580528879, −2.44067669426493331013866207385, −2.41988204609080166936116721467, −2.40817625766789171744055404402, −2.15679729601888611495641418221, −2.06195813260689456489757701098, −1.81644633450719847960128541626, −1.80530541712926962042146616910, −1.66098590292052377467303547989, −1.59483744892557199357530624630, −1.43359672929669549181686826480, −1.31754909456981917226065481042, −1.17591020514680373916880319703, −0.874002945697042183210442634176, −0.855975853610935063396770596369, −0.60125364901609698737795162697, −0.58581048014131220635449908050, −0.53115451270492481860071290342, −0.48985967170940796626271900410, −0.06462853463235260838480154248,
0.06462853463235260838480154248, 0.48985967170940796626271900410, 0.53115451270492481860071290342, 0.58581048014131220635449908050, 0.60125364901609698737795162697, 0.855975853610935063396770596369, 0.874002945697042183210442634176, 1.17591020514680373916880319703, 1.31754909456981917226065481042, 1.43359672929669549181686826480, 1.59483744892557199357530624630, 1.66098590292052377467303547989, 1.80530541712926962042146616910, 1.81644633450719847960128541626, 2.06195813260689456489757701098, 2.15679729601888611495641418221, 2.40817625766789171744055404402, 2.41988204609080166936116721467, 2.44067669426493331013866207385, 2.49674757074081814029580528879, 2.63699848720636296384487113238, 2.68129724530052194972945765940, 2.73848262351091451067122265452, 2.91224626360531064333961719534, 3.22642728640231108350096066403
Plot not available for L-functions of degree greater than 10.