| L(s) = 1 | + 4·2-s + 8·4-s + 38·7-s − 404·11-s + 198·13-s + 152·14-s − 64·16-s − 478·17-s − 1.61e3·22-s + 1.08e3·23-s + 792·26-s + 304·28-s − 1.51e3·31-s − 256·32-s − 1.91e3·34-s − 282·37-s − 2.08e3·41-s + 1.51e3·43-s − 3.23e3·44-s + 4.32e3·46-s − 918·47-s + 722·49-s + 1.58e3·52-s − 3.63e3·53-s + 4.16e3·61-s − 6.06e3·62-s − 512·64-s + ⋯ |
| L(s) = 1 | + 2-s + 1/2·4-s + 0.775·7-s − 3.33·11-s + 1.17·13-s + 0.775·14-s − 1/4·16-s − 1.65·17-s − 3.33·22-s + 2.04·23-s + 1.17·26-s + 0.387·28-s − 1.57·31-s − 1/4·32-s − 1.65·34-s − 0.205·37-s − 1.23·41-s + 0.820·43-s − 1.66·44-s + 2.04·46-s − 0.415·47-s + 0.300·49-s + 0.585·52-s − 1.29·53-s + 1.11·61-s − 1.57·62-s − 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.386873037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.386873037\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 38 T + 722 T^{2} - 38 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 202 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 198 T + 19602 T^{2} - 198 p^{4} T^{3} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 478 T + 114242 T^{2} + 478 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 259042 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1082 T + 585362 T^{2} - 1082 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1374562 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 758 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 282 T + 39762 T^{2} + 282 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 1042 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1518 T + 1152162 T^{2} - 1518 p^{4} T^{3} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 918 T + 421362 T^{2} + 918 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3638 T + 6617522 T^{2} + 3638 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3074722 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2082 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10162 T + 51633122 T^{2} + 10162 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3478 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6958 T + 24206882 T^{2} - 6958 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 18917762 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12162 T + 73957122 T^{2} - 12162 p^{4} T^{3} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 93222082 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1122 T + 629442 T^{2} + 1122 p^{4} T^{3} + p^{8} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96999253601733227639599294480, −10.54178138724810272434608563139, −10.01396631993002159022651932297, −9.145791811846195301443084102845, −8.922371224658770934622585117583, −8.313932747718202344688824671125, −7.937788245208023242520178803721, −7.53023156422885956153119614827, −6.92425026304366330855943559151, −6.49063078554468423026847434132, −5.72138955222983763643924306937, −5.32572834554432456518347281689, −4.83738942617845093871461634636, −4.82508294770116238484836714126, −3.80570161603564770186637160346, −3.28123628879066637036403378974, −2.57544236264127434810592184589, −2.26822528518633115307415035659, −1.37810788511410099850060729833, −0.25741579218555064151217511903,
0.25741579218555064151217511903, 1.37810788511410099850060729833, 2.26822528518633115307415035659, 2.57544236264127434810592184589, 3.28123628879066637036403378974, 3.80570161603564770186637160346, 4.82508294770116238484836714126, 4.83738942617845093871461634636, 5.32572834554432456518347281689, 5.72138955222983763643924306937, 6.49063078554468423026847434132, 6.92425026304366330855943559151, 7.53023156422885956153119614827, 7.937788245208023242520178803721, 8.313932747718202344688824671125, 8.922371224658770934622585117583, 9.145791811846195301443084102845, 10.01396631993002159022651932297, 10.54178138724810272434608563139, 10.96999253601733227639599294480
