| L(s) = 1 | − 4·4-s − 9·9-s + 112·11-s + 16·16-s − 72·19-s + 516·29-s − 80·31-s + 36·36-s − 292·41-s − 448·44-s − 49·49-s − 376·59-s + 188·61-s − 64·64-s + 1.06e3·71-s + 288·76-s + 1.07e3·79-s + 81·81-s + 2.18e3·89-s − 1.00e3·99-s − 1.54e3·101-s + 1.42e3·109-s − 2.06e3·116-s + 6.74e3·121-s + 320·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 3.06·11-s + 1/4·16-s − 0.869·19-s + 3.30·29-s − 0.463·31-s + 1/6·36-s − 1.11·41-s − 1.53·44-s − 1/7·49-s − 0.829·59-s + 0.394·61-s − 1/8·64-s + 1.77·71-s + 0.434·76-s + 1.52·79-s + 1/9·81-s + 2.59·89-s − 1.02·99-s − 1.52·101-s + 1.25·109-s − 1.65·116-s + 5.06·121-s + 0.231·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.978530462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.978530462\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1478 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 104 T + p^{3} T^{2} )( 1 + 104 T + p^{3} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17278 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 258 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 40 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 69622 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 146 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 137110 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 167646 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 280854 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 188 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 94 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 404390 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 532 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 185134 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 536 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 14202 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1090 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 202270 T^{2} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.537867678191356810747928358869, −9.254509599157413279723815813687, −9.038999977671782545977725702392, −8.431540335497748858638449259711, −8.344015855028004026109504446417, −7.82762633058660843569613066908, −6.89003438673312098780383388300, −6.81915867519050058659768398753, −6.29255979644036465093833985892, −6.25089113615143327942675766899, −5.40260194828026943495146703605, −4.85986534764098898693266491283, −4.34784906589297743585541279576, −4.17923162200247671281942043273, −3.35570519854764500916905122191, −3.29027775667560680077691223501, −2.23717599392631223216453485328, −1.69924617690997853704477426626, −0.969642020677973194411448156178, −0.62758468262796409261385898766,
0.62758468262796409261385898766, 0.969642020677973194411448156178, 1.69924617690997853704477426626, 2.23717599392631223216453485328, 3.29027775667560680077691223501, 3.35570519854764500916905122191, 4.17923162200247671281942043273, 4.34784906589297743585541279576, 4.85986534764098898693266491283, 5.40260194828026943495146703605, 6.25089113615143327942675766899, 6.29255979644036465093833985892, 6.81915867519050058659768398753, 6.89003438673312098780383388300, 7.82762633058660843569613066908, 8.344015855028004026109504446417, 8.431540335497748858638449259711, 9.038999977671782545977725702392, 9.254509599157413279723815813687, 9.537867678191356810747928358869
