| L(s) = 1 | − 2·4-s + 12·7-s − 12·13-s + 3·16-s + 6·17-s + 8·19-s + 25-s − 24·28-s + 4·31-s + 12·37-s + 12·41-s − 10·43-s + 24·47-s + 62·49-s + 24·52-s − 12·53-s + 12·59-s + 8·61-s − 4·64-s − 12·68-s + 18·71-s − 16·76-s − 12·83-s + 12·89-s − 144·91-s − 2·100-s + 8·103-s + ⋯ |
| L(s) = 1 | − 4-s + 4.53·7-s − 3.32·13-s + 3/4·16-s + 1.45·17-s + 1.83·19-s + 1/5·25-s − 4.53·28-s + 0.718·31-s + 1.97·37-s + 1.87·41-s − 1.52·43-s + 3.50·47-s + 62/7·49-s + 3.32·52-s − 1.64·53-s + 1.56·59-s + 1.02·61-s − 1/2·64-s − 1.45·68-s + 2.13·71-s − 1.83·76-s − 1.31·83-s + 1.27·89-s − 15.0·91-s − 1/5·100-s + 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.398230646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.398230646\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 96 T^{3} + 511 T^{4} - 96 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 482 T^{4} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2730 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2722 T^{4} - 444 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 50 T^{2} + 60 T^{3} - 1297 T^{4} + 60 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 119 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 70 T^{2} + p^{2} T^{4} ) \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 6034 T^{4} - 660 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 822 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18 T + 162 T^{2} - 1908 T^{3} + 21247 T^{4} - 1908 p T^{5} + 162 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 11418 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 164 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 1020 T^{3} + 14434 T^{4} - 1020 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 27258 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.24715923921702683882828561560, −7.09780921840605626945989537599, −6.40421057616872906440436249138, −6.40398319397851394612348190114, −5.98683033631751049715636533739, −5.51555217751654097820118157861, −5.43892374124143374491050220239, −5.34027496549428311589585850124, −5.16745575636973977070069412519, −4.91464255799920536244612410552, −4.81819266512938655188647853956, −4.58687499527239608046961501382, −4.37160883392010355454315968174, −4.18525462961676026140668310225, −3.95429649371921879594383759264, −3.45897407262338544297935908542, −3.17044570193866350115444339721, −2.82340255985835075528964904137, −2.31630297723477979310650756355, −2.25032375592871036513916906282, −2.23663052533078633366857090032, −1.49380732138562724007481574500, −1.22846258803776149481102873134, −0.997134463387902239702962421724, −0.62879517804825981032296192473,
0.62879517804825981032296192473, 0.997134463387902239702962421724, 1.22846258803776149481102873134, 1.49380732138562724007481574500, 2.23663052533078633366857090032, 2.25032375592871036513916906282, 2.31630297723477979310650756355, 2.82340255985835075528964904137, 3.17044570193866350115444339721, 3.45897407262338544297935908542, 3.95429649371921879594383759264, 4.18525462961676026140668310225, 4.37160883392010355454315968174, 4.58687499527239608046961501382, 4.81819266512938655188647853956, 4.91464255799920536244612410552, 5.16745575636973977070069412519, 5.34027496549428311589585850124, 5.43892374124143374491050220239, 5.51555217751654097820118157861, 5.98683033631751049715636533739, 6.40398319397851394612348190114, 6.40421057616872906440436249138, 7.09780921840605626945989537599, 7.24715923921702683882828561560
