L(s) = 1 | + (−0.266 + 0.223i)3-s + (−0.152 + 0.866i)9-s + (0.766 + 1.32i)11-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)19-s + (0.766 + 0.642i)25-s + (−0.326 − 0.565i)27-s + (−0.5 − 0.181i)33-s + (−1.43 + 1.20i)41-s + (−0.939 − 0.342i)43-s + (−0.5 − 0.866i)49-s + (0.266 + 0.223i)51-s + (−0.173 + 0.300i)57-s + (−0.266 − 1.50i)59-s + (0.326 − 1.85i)67-s + ⋯ |
L(s) = 1 | + (−0.266 + 0.223i)3-s + (−0.152 + 0.866i)9-s + (0.766 + 1.32i)11-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)19-s + (0.766 + 0.642i)25-s + (−0.326 − 0.565i)27-s + (−0.5 − 0.181i)33-s + (−1.43 + 1.20i)41-s + (−0.939 − 0.342i)43-s + (−0.5 − 0.866i)49-s + (0.266 + 0.223i)51-s + (−0.173 + 0.300i)57-s + (−0.266 − 1.50i)59-s + (0.326 − 1.85i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8640133113\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8640133113\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-0.939 + 0.342i)T \) |
good | 3 | \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 5 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 83 | \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
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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
−11.08294334497560413895395610556, −9.921827146520219298617441610317, −9.479172007547607425185062254661, −8.318044661630414524699344578636, −7.29498620924618147029399693010, −6.63185417040064182495879095091, −5.13954695439333345400622437016, −4.73310911845416777839096337163, −3.25986610755898420779037744957, −1.83779805343509708672607882732,
1.22935428165334073002954750685, 3.13031585724353341521928791601, 3.99397443482827874161638752104, 5.48808465552932824849095915287, 6.24584583255770055255394976602, 7.00064296378201153496216777063, 8.331051208528418262941421363205, 8.876364346904785175355093423578, 9.905488152862682249298056440472, 10.86513780020610467402695864701