| L(s) = 1 | + 16·4-s − 76·7-s + 36·9-s + 360·11-s + 192·16-s − 792·23-s + 100·25-s − 1.21e3·28-s + 1.22e3·29-s + 576·36-s − 3.89e3·37-s + 3.68e3·43-s + 5.76e3·44-s + 2.12e3·49-s + 5.83e3·53-s − 2.73e3·63-s + 2.04e3·64-s − 1.04e3·67-s − 2.15e4·71-s − 2.73e4·77-s + 1.27e4·79-s − 1.18e4·81-s − 1.26e4·92-s + 1.29e4·99-s + 1.60e3·100-s − 1.00e4·107-s − 1.34e4·109-s + ⋯ |
| L(s) = 1 | + 4-s − 1.55·7-s + 4/9·9-s + 2.97·11-s + 3/4·16-s − 1.49·23-s + 4/25·25-s − 1.55·28-s + 1.45·29-s + 4/9·36-s − 2.84·37-s + 1.99·43-s + 2.97·44-s + 0.883·49-s + 2.07·53-s − 0.689·63-s + 1/2·64-s − 0.233·67-s − 4.27·71-s − 4.61·77-s + 2.04·79-s − 1.80·81-s − 1.49·92-s + 1.32·99-s + 4/25·100-s − 0.874·107-s − 1.13·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.787721713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.787721713\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 76 T + 522 p T^{2} + 76 p^{4} T^{3} + p^{8} T^{4} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 4 p^{2} T^{2} + 1462 p^{2} T^{4} - 4 p^{10} T^{6} + p^{16} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 4 p^{2} T^{2} + 345702 T^{4} - 4 p^{10} T^{6} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 180 T + 27014 T^{2} - 180 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 111460 T^{2} + 4736358630 T^{4} - 111460 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 217348 T^{2} + 25123879686 T^{4} - 217348 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 150436 T^{2} + 9183572838 T^{4} - 150436 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 396 T + 525158 T^{2} + 396 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 612 T + 1092326 T^{2} - 612 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 1703428 T^{2} + 1577900136966 T^{4} - 1703428 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 1948 T + 4603686 T^{2} + 1948 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 7851268 T^{2} + 31379741059206 T^{4} - 7851268 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 1844 T + 6650886 T^{2} - 1844 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 10858756 T^{2} + 58650967963398 T^{4} - 10858756 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 2916 T + 9386534 T^{2} - 2916 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 42750244 T^{2} + 750538168343526 T^{4} - 42750244 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 28277476 T^{2} + 401277478909158 T^{4} - 28277476 p^{8} T^{6} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 524 T + 39707334 T^{2} + 524 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 10764 T + 70144454 T^{2} + 10764 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 62983684 T^{2} + 1969418053172358 T^{4} - 62983684 p^{8} T^{6} + p^{16} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 6388 T + 73521798 T^{2} - 6388 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 129024292 T^{2} + 7865129580692070 T^{4} - 129024292 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 228854020 T^{2} + 20924407354852230 T^{4} - 228854020 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 150468100 T^{2} + 16842128301249030 T^{4} - 150468100 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−14.25460658063141576353703847217, −14.20327862472762734726708114811, −13.40110562771398620534322710302, −13.23293435735272338433698154774, −12.51472534075513744428891950515, −12.18627026451762225964163807564, −11.93294322758369048840350546679, −11.87669990326372847463667711754, −11.32932679614669470779169211967, −10.39586744061276120199015969284, −10.39234639683878339141579667076, −10.10888711979164478437958395981, −9.222831216604606739653748821469, −9.151420707955181608830999583110, −8.764331441211042225167862507396, −7.972395856919718186848184736214, −7.21662346057119547636785436441, −6.86385157534129198436318472719, −6.54951202933580847798732327619, −6.16485553035341295092316063267, −5.55734086118024202494047993454, −4.10267712144115808830696036155, −3.93226243411066662269045761385, −2.92183054637341677873953587233, −1.48434266230414316686210017856,
1.48434266230414316686210017856, 2.92183054637341677873953587233, 3.93226243411066662269045761385, 4.10267712144115808830696036155, 5.55734086118024202494047993454, 6.16485553035341295092316063267, 6.54951202933580847798732327619, 6.86385157534129198436318472719, 7.21662346057119547636785436441, 7.972395856919718186848184736214, 8.764331441211042225167862507396, 9.151420707955181608830999583110, 9.222831216604606739653748821469, 10.10888711979164478437958395981, 10.39234639683878339141579667076, 10.39586744061276120199015969284, 11.32932679614669470779169211967, 11.87669990326372847463667711754, 11.93294322758369048840350546679, 12.18627026451762225964163807564, 12.51472534075513744428891950515, 13.23293435735272338433698154774, 13.40110562771398620534322710302, 14.20327862472762734726708114811, 14.25460658063141576353703847217
