| L(s) = 1 | − 1.56·2-s + 3-s + 0.438·4-s + 5-s − 1.56·6-s + 5.12·7-s + 2.43·8-s + 9-s − 1.56·10-s − 1.43·11-s + 0.438·12-s − 2·13-s − 8·14-s + 15-s − 4.68·16-s − 7.12·17-s − 1.56·18-s + 5.12·19-s + 0.438·20-s + 5.12·21-s + 2.24·22-s + 6.56·23-s + 2.43·24-s + 25-s + 3.12·26-s + 27-s + 2.24·28-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.577·3-s + 0.219·4-s + 0.447·5-s − 0.637·6-s + 1.93·7-s + 0.862·8-s + 0.333·9-s − 0.493·10-s − 0.433·11-s + 0.126·12-s − 0.554·13-s − 2.13·14-s + 0.258·15-s − 1.17·16-s − 1.72·17-s − 0.368·18-s + 1.17·19-s + 0.0980·20-s + 1.11·21-s + 0.478·22-s + 1.36·23-s + 0.497·24-s + 0.200·25-s + 0.612·26-s + 0.192·27-s + 0.424·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.160201666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160201666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 3.43T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 2.87T + 71T^{2} \) |
| 73 | \( 1 - 1.68T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 2.56T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 5.68T + 97T^{2} \) |
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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
−10.96417206488600490263655285433, −10.10904995326953672334215539315, −9.118052207317669976865804380000, −8.543204798214796364896863540620, −7.75235415881385980632209288744, −7.00095725827243836204529869390, −5.11028811136306908507362470100, −4.54100118367316077581297147597, −2.46123361879166738208870635262, −1.36360332662919032380159427521,
1.36360332662919032380159427521, 2.46123361879166738208870635262, 4.54100118367316077581297147597, 5.11028811136306908507362470100, 7.00095725827243836204529869390, 7.75235415881385980632209288744, 8.543204798214796364896863540620, 9.118052207317669976865804380000, 10.10904995326953672334215539315, 10.96417206488600490263655285433
