L(s) = 1 | + (0.198 + 0.0646i)2-s + (−0.773 − 0.562i)4-s + (−0.587 + 0.809i)7-s + (−0.240 − 0.330i)8-s + (−0.309 + 0.951i)9-s + (0.669 − 0.743i)11-s + (−0.169 + 0.122i)14-s + (0.269 + 0.828i)16-s + (−0.122 + 0.169i)18-s + (0.181 − 0.104i)22-s + 1.82i·23-s + (0.909 − 0.295i)28-s + (1.08 + 0.786i)29-s + 0.591i·32-s + (0.773 − 0.562i)36-s + (0.122 − 0.169i)37-s + ⋯ |
L(s) = 1 | + (0.198 + 0.0646i)2-s + (−0.773 − 0.562i)4-s + (−0.587 + 0.809i)7-s + (−0.240 − 0.330i)8-s + (−0.309 + 0.951i)9-s + (0.669 − 0.743i)11-s + (−0.169 + 0.122i)14-s + (0.269 + 0.828i)16-s + (−0.122 + 0.169i)18-s + (0.181 − 0.104i)22-s + 1.82i·23-s + (0.909 − 0.295i)28-s + (1.08 + 0.786i)29-s + 0.591i·32-s + (0.773 − 0.562i)36-s + (0.122 − 0.169i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8792844607\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8792844607\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (0.587 - 0.809i)T \) |
| 11 | \( 1 + (-0.669 + 0.743i)T \) |
good | 2 | \( 1 + (-0.198 - 0.0646i)T + (0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - 1.82iT - T^{2} \) |
| 29 | \( 1 + (-1.08 - 0.786i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.122 + 0.169i)T + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - 1.33iT - T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - 1.95iT - T^{2} \) |
| 71 | \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.604 + 1.86i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.472117122158326907853625062363, −8.774412438462872102548156985823, −8.215520911855402830929187212531, −7.06327385231331898663213444997, −6.04625680265722287079801895403, −5.58794035658278504016411438086, −4.79990855637797752916535910396, −3.71964849759041430748556537511, −2.81717550655938838123495958033, −1.39556226581192290970570505601,
0.67557532892651238843157288527, 2.56494106290740398511844325094, 3.61761555929739279722903931553, 4.17222988241712658431952976575, 4.96899314254969428091012642950, 6.31487932982232760291827582631, 6.74342719731948375050232462587, 7.74319281504669500854283618916, 8.589945510889122568456453547020, 9.245708538464342015500863164242