| L(s) = 1 | − 0.470·2-s − 2.24·3-s − 1.77·4-s + 1.05·6-s + 7-s + 1.77·8-s + 2.05·9-s − 2.24·11-s + 4.00·12-s − 13-s − 0.470·14-s + 2.71·16-s + 1.30·17-s − 0.968·18-s − 1.47·19-s − 2.24·21-s + 1.05·22-s − 5.83·23-s − 4.00·24-s + 0.470·26-s + 2.11·27-s − 1.77·28-s + 5.22·29-s − 7.02·31-s − 4.83·32-s + 5.05·33-s − 0.615·34-s + ⋯ |
| L(s) = 1 | − 0.332·2-s − 1.29·3-s − 0.889·4-s + 0.432·6-s + 0.377·7-s + 0.628·8-s + 0.686·9-s − 0.678·11-s + 1.15·12-s − 0.277·13-s − 0.125·14-s + 0.679·16-s + 0.317·17-s − 0.228·18-s − 0.337·19-s − 0.490·21-s + 0.225·22-s − 1.21·23-s − 0.816·24-s + 0.0923·26-s + 0.407·27-s − 0.336·28-s + 0.969·29-s − 1.26·31-s − 0.855·32-s + 0.880·33-s − 0.105·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4169169005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4169169005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.470T + 2T^{2} \) |
| 3 | \( 1 + 2.24T + 3T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 - 5.22T + 29T^{2} \) |
| 31 | \( 1 + 7.02T + 31T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 + 7.64T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 6.91T + 89T^{2} \) |
| 97 | \( 1 - 3.47T + 97T^{2} \) |
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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
−9.076928555227673373343212537417, −8.122561416159479436814178082714, −7.69709778193808887446665095622, −6.51941354111180638329477221173, −5.80051425466418093098220704901, −4.97710521023129581095395500029, −4.58490952603226215690408259229, −3.37491041055556279617053478705, −1.80622807390784857096023020607, −0.47157078584385429360182253201,
0.47157078584385429360182253201, 1.80622807390784857096023020607, 3.37491041055556279617053478705, 4.58490952603226215690408259229, 4.97710521023129581095395500029, 5.80051425466418093098220704901, 6.51941354111180638329477221173, 7.69709778193808887446665095622, 8.122561416159479436814178082714, 9.076928555227673373343212537417
