L(s) = 1 | + (0.932 + 0.361i)2-s + (0.739 + 0.673i)4-s + (0.397 − 0.798i)5-s + (0.445 + 0.895i)8-s + (−0.273 − 0.961i)9-s + (0.658 − 0.600i)10-s + (−0.243 − 0.489i)13-s + (0.0922 + 0.995i)16-s + (0.0922 + 0.995i)17-s + (0.0922 − 0.995i)18-s + (0.831 − 0.322i)20-s + (0.123 + 0.163i)25-s + (−0.0505 − 0.544i)26-s + (−0.404 − 0.368i)29-s + (−0.273 + 0.961i)32-s + ⋯ |
L(s) = 1 | + (0.932 + 0.361i)2-s + (0.739 + 0.673i)4-s + (0.397 − 0.798i)5-s + (0.445 + 0.895i)8-s + (−0.273 − 0.961i)9-s + (0.658 − 0.600i)10-s + (−0.243 − 0.489i)13-s + (0.0922 + 0.995i)16-s + (0.0922 + 0.995i)17-s + (0.0922 − 0.995i)18-s + (0.831 − 0.322i)20-s + (0.123 + 0.163i)25-s + (−0.0505 − 0.544i)26-s + (−0.404 − 0.368i)29-s + (−0.273 + 0.961i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.876954343\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.876954343\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.932 - 0.361i)T \) |
| 17 | \( 1 + (-0.0922 - 0.995i)T \) |
good | 3 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 5 | \( 1 + (-0.397 + 0.798i)T + (-0.602 - 0.798i)T^{2} \) |
| 7 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 11 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 13 | \( 1 + (0.243 + 0.489i)T + (-0.602 + 0.798i)T^{2} \) |
| 19 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 23 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 29 | \( 1 + (0.404 + 0.368i)T + (0.0922 + 0.995i)T^{2} \) |
| 31 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 37 | \( 1 + (1.83 + 0.342i)T + (0.932 + 0.361i)T^{2} \) |
| 41 | \( 1 + (0.537 - 0.711i)T + (-0.273 - 0.961i)T^{2} \) |
| 43 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 47 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 53 | \( 1 + (-0.465 - 1.63i)T + (-0.850 + 0.526i)T^{2} \) |
| 59 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 61 | \( 1 + (-1.02 + 0.634i)T + (0.445 - 0.895i)T^{2} \) |
| 67 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 71 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 73 | \( 1 + (0.0505 + 0.544i)T + (-0.982 + 0.183i)T^{2} \) |
| 79 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 83 | \( 1 + (0.273 - 0.961i)T^{2} \) |
| 89 | \( 1 + (-0.831 + 1.66i)T + (-0.602 - 0.798i)T^{2} \) |
| 97 | \( 1 + (0.404 + 1.42i)T + (-0.850 + 0.526i)T^{2} \) |
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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
−10.02576455001162777988856303726, −9.003847896923087837545160013800, −8.377545859125564949811489542264, −7.41049173208504637475188594068, −6.43107381543176030387176519927, −5.73323825440418030526315591180, −4.99059495217311847748369296939, −3.95057420323977737426872435840, −3.07375947480542724242307597521, −1.64028543105429784179796503169,
1.92048349028876229258633561738, 2.71547983211492471513436241733, 3.71529688182733058189612833105, 4.98137871149347705318385217063, 5.44901632063214513737841555527, 6.73304006883648306222873089043, 7.04857651971487567854503045679, 8.246417730016807730480404496638, 9.435498069429516014252589506344, 10.26503664583973994359755726551