L(s) = 1 | + (−0.00918 − 0.999i)2-s + (1.75 + 0.532i)3-s + (−0.999 + 0.0183i)4-s + (0.516 − 1.76i)6-s + (0.0275 + 0.999i)8-s + (1.98 + 1.32i)9-s + (−1.10 + 1.50i)11-s + (−1.76 − 0.500i)12-s + (0.999 − 0.0367i)16-s + (0.132 + 0.240i)17-s + (1.30 − 1.99i)18-s + (0.870 − 0.492i)19-s + (1.51 + 1.09i)22-s + (−0.484 + 1.77i)24-s + (−0.435 + 0.900i)25-s + ⋯ |
L(s) = 1 | + (−0.00918 − 0.999i)2-s + (1.75 + 0.532i)3-s + (−0.999 + 0.0183i)4-s + (0.516 − 1.76i)6-s + (0.0275 + 0.999i)8-s + (1.98 + 1.32i)9-s + (−1.10 + 1.50i)11-s + (−1.76 − 0.500i)12-s + (0.999 − 0.0367i)16-s + (0.132 + 0.240i)17-s + (1.30 − 1.99i)18-s + (0.870 − 0.492i)19-s + (1.51 + 1.09i)22-s + (−0.484 + 1.77i)24-s + (−0.435 + 0.900i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.963188425\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.963188425\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.00918 + 0.999i)T \) |
| 19 | \( 1 + (-0.870 + 0.492i)T \) |
good | 3 | \( 1 + (-1.75 - 0.532i)T + (0.832 + 0.554i)T^{2} \) |
| 5 | \( 1 + (0.435 - 0.900i)T^{2} \) |
| 7 | \( 1 + (0.926 + 0.376i)T^{2} \) |
| 11 | \( 1 + (1.10 - 1.50i)T + (-0.298 - 0.954i)T^{2} \) |
| 13 | \( 1 + (0.971 + 0.236i)T^{2} \) |
| 17 | \( 1 + (-0.132 - 0.240i)T + (-0.531 + 0.847i)T^{2} \) |
| 23 | \( 1 + (0.896 + 0.443i)T^{2} \) |
| 29 | \( 1 + (0.951 - 0.307i)T^{2} \) |
| 31 | \( 1 + (0.754 + 0.656i)T^{2} \) |
| 37 | \( 1 + (0.677 - 0.735i)T^{2} \) |
| 41 | \( 1 + (0.924 + 1.30i)T + (-0.333 + 0.942i)T^{2} \) |
| 43 | \( 1 + (-1.12 + 0.584i)T + (0.577 - 0.816i)T^{2} \) |
| 47 | \( 1 + (-0.888 + 0.459i)T^{2} \) |
| 53 | \( 1 + (0.0459 - 0.998i)T^{2} \) |
| 59 | \( 1 + (-1.29 + 0.119i)T + (0.983 - 0.182i)T^{2} \) |
| 61 | \( 1 + (0.119 + 0.992i)T^{2} \) |
| 67 | \( 1 + (0.0798 - 0.100i)T + (-0.227 - 0.973i)T^{2} \) |
| 71 | \( 1 + (0.119 - 0.992i)T^{2} \) |
| 73 | \( 1 + (-0.595 + 0.989i)T + (-0.467 - 0.883i)T^{2} \) |
| 79 | \( 1 + (0.995 + 0.0917i)T^{2} \) |
| 83 | \( 1 + (0.261 + 1.88i)T + (-0.962 + 0.272i)T^{2} \) |
| 89 | \( 1 + (0.955 + 1.58i)T + (-0.467 + 0.883i)T^{2} \) |
| 97 | \( 1 + (-1.16 - 1.47i)T + (-0.227 + 0.973i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.141142830846376145380818842111, −8.403593959982768472356762145675, −7.63606267047770318952267033594, −7.21937898524047691624950958722, −5.35471818955013677441808806451, −4.76549809689472680361907148155, −3.90989786660171834908566848589, −3.21034394847814771717102601567, −2.37215099463410916207310473635, −1.76263991542528062447062777695,
1.10788722447464078896026536603, 2.64036115970928423336041799177, 3.28322850883167015991250600819, 4.10107737141079392898060909920, 5.24198015893015617040840131651, 6.08105018530287749064807405901, 6.92447034448421722167390867466, 7.71899584076553226225628532179, 8.267771852384874332084712725118, 8.438919973047126092269798171699