L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)6-s + (1 − 1.73i)7-s − 8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (0.999 + 1.73i)14-s + (0.5 − 0.866i)16-s + 0.999·18-s + 19-s + (0.999 + 1.73i)21-s + (0.499 + 0.866i)22-s + (0.5 + 0.866i)23-s + (0.499 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)6-s + (1 − 1.73i)7-s − 8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (0.999 + 1.73i)14-s + (0.5 − 0.866i)16-s + 0.999·18-s + 19-s + (0.999 + 1.73i)21-s + (0.499 + 0.866i)22-s + (0.5 + 0.866i)23-s + (0.499 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8210286888\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8210286888\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - 2T + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 2T + T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.938889609463993239076139993217, −9.091281674116506414438019337703, −8.465570964051848098140038582638, −7.41241232742457990445604674169, −6.99767094037660796518173280472, −5.94357438983016738342417216607, −5.12200633924329992388128251392, −3.95522946833482582291930552116, −3.47480962363291170688952117094, −1.13847392226770135971378215185,
1.26394259269510389978642976286, 2.16051852425647390494265817457, 2.90604918736152011723007270213, 4.77412569709562646225241726885, 5.64896942522348988063839843898, 6.17263016718159237715365301768, 7.31860851631546025744466348630, 8.375956581134059663159555785259, 8.752514260786674733772215146248, 9.794462920488303623500360124282