| L(s) = 1 | + 0.414·2-s − 1.82·4-s − 5-s − 7-s − 1.58·8-s − 0.414·10-s − 4.82·11-s + 0.828·13-s − 0.414·14-s + 3·16-s − 7.65·17-s − 2.82·19-s + 1.82·20-s − 1.99·22-s + 3.65·23-s + 25-s + 0.343·26-s + 1.82·28-s − 8·29-s + 8.48·31-s + 4.41·32-s − 3.17·34-s + 35-s − 6·37-s − 1.17·38-s + 1.58·40-s + 7.65·41-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 0.914·4-s − 0.447·5-s − 0.377·7-s − 0.560·8-s − 0.130·10-s − 1.45·11-s + 0.229·13-s − 0.110·14-s + 0.750·16-s − 1.85·17-s − 0.648·19-s + 0.408·20-s − 0.426·22-s + 0.762·23-s + 0.200·25-s + 0.0672·26-s + 0.345·28-s − 1.48·29-s + 1.52·31-s + 0.780·32-s − 0.543·34-s + 0.169·35-s − 0.986·37-s − 0.190·38-s + 0.250·40-s + 1.19·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 5.17T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 5.31T + 89T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
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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
−11.11523555624672312666832883670, −10.36275811704857005484562393732, −9.154078141054155021048351273996, −8.488643266238240589053142416595, −7.40320449581073324832387392396, −6.12522467857803522401585965672, −4.96352485699437409756559981220, −4.08738209618549570146385865450, −2.71804868450455840630447744204, 0,
2.71804868450455840630447744204, 4.08738209618549570146385865450, 4.96352485699437409756559981220, 6.12522467857803522401585965672, 7.40320449581073324832387392396, 8.488643266238240589053142416595, 9.154078141054155021048351273996, 10.36275811704857005484562393732, 11.11523555624672312666832883670
