| L(s) = 1 | + 6·9-s − 4·19-s + 8·25-s − 4·29-s − 8·31-s + 22·49-s + 52·59-s + 8·61-s + 16·71-s + 8·79-s + 17·81-s + 32·89-s + 24·101-s + 20·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s − 24·171-s + 173-s + ⋯ |
| L(s) = 1 | + 2·9-s − 0.917·19-s + 8/5·25-s − 0.742·29-s − 1.43·31-s + 22/7·49-s + 6.76·59-s + 1.02·61-s + 1.89·71-s + 0.900·79-s + 17/9·81-s + 3.39·89-s + 2.38·101-s + 1.91·109-s − 3.63·121-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 + 2.61·169-s − 1.83·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.554221467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.554221467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 211 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 595 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 54 T^{2} + 1715 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 64 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 694 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7506 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 194 T^{2} + 14995 T^{4} - 194 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 26 T + 285 T^{2} - 26 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 124 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 166 T^{2} + 15667 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 7075 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6774 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 16 T + 224 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 316 T^{2} + 43270 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.81818591236742531507530799066, −6.53847824451944078476942820375, −6.31443354785152808386557464951, −6.31017275991180559942611282348, −6.02828787343619651952209657033, −5.38197193791193878841825432983, −5.30627092516066515155412615147, −5.17451060186451044927248202297, −5.11127440523543769365541882968, −5.06687546685037456833481733925, −4.27739640598052615362890623464, −4.25801468238873508664717172334, −4.00902587571035455570869971860, −3.80605261575253879895754112899, −3.67564156476820274354921270460, −3.65674990583320570145715394290, −2.98928491061079411314799678318, −2.55503064524296876872992391507, −2.52657008241130813305408273596, −2.12888077900738100818390995859, −1.91332638754287278901993752509, −1.74327967522851569861427311134, −0.916294492091942959435409000840, −0.900047003055115815618466990919, −0.63338995933067839325469605499,
0.63338995933067839325469605499, 0.900047003055115815618466990919, 0.916294492091942959435409000840, 1.74327967522851569861427311134, 1.91332638754287278901993752509, 2.12888077900738100818390995859, 2.52657008241130813305408273596, 2.55503064524296876872992391507, 2.98928491061079411314799678318, 3.65674990583320570145715394290, 3.67564156476820274354921270460, 3.80605261575253879895754112899, 4.00902587571035455570869971860, 4.25801468238873508664717172334, 4.27739640598052615362890623464, 5.06687546685037456833481733925, 5.11127440523543769365541882968, 5.17451060186451044927248202297, 5.30627092516066515155412615147, 5.38197193791193878841825432983, 6.02828787343619651952209657033, 6.31017275991180559942611282348, 6.31443354785152808386557464951, 6.53847824451944078476942820375, 6.81818591236742531507530799066
