| L(s) = 1 | + 0.414·2-s + 3-s − 1.82·4-s + 0.414·6-s + 2.41·7-s − 1.58·8-s + 9-s − 1.82·12-s − 2.82·13-s + 0.999·14-s + 3·16-s + 0.414·17-s + 0.414·18-s − 3.58·19-s + 2.41·21-s + 23-s − 1.58·24-s − 1.17·26-s + 27-s − 4.41·28-s − 6.82·29-s + 8.48·31-s + 4.41·32-s + 0.171·34-s − 1.82·36-s − 5.82·37-s − 1.48·38-s + ⋯ |
| L(s) = 1 | + 0.292·2-s + 0.577·3-s − 0.914·4-s + 0.169·6-s + 0.912·7-s − 0.560·8-s + 0.333·9-s − 0.527·12-s − 0.784·13-s + 0.267·14-s + 0.750·16-s + 0.100·17-s + 0.0976·18-s − 0.822·19-s + 0.526·21-s + 0.208·23-s − 0.323·24-s − 0.229·26-s + 0.192·27-s − 0.834·28-s − 1.26·29-s + 1.52·31-s + 0.780·32-s + 0.0294·34-s − 0.304·36-s − 0.958·37-s − 0.240·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 0.414T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 5.82T + 37T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 - 0.343T + 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 - 11T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 2.17T + 71T^{2} \) |
| 73 | \( 1 - 3.17T + 73T^{2} \) |
| 79 | \( 1 + 4.75T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 - 0.171T + 97T^{2} \) |
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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
−7.50469309451893261438231895111, −6.81298432461270436620901541841, −5.82479161882934493505969910793, −5.13703294359125668727380465378, −4.56693246416548667666418615208, −3.98324501866706610597225058629, −3.13261525236138259243313068159, −2.25892544466979093178355432280, −1.33038809008974848203611075154, 0,
1.33038809008974848203611075154, 2.25892544466979093178355432280, 3.13261525236138259243313068159, 3.98324501866706610597225058629, 4.56693246416548667666418615208, 5.13703294359125668727380465378, 5.82479161882934493505969910793, 6.81298432461270436620901541841, 7.50469309451893261438231895111
