| L(s) = 1 | + 2-s − 1.59·3-s + 4-s + 3.74·5-s − 1.59·6-s − 7-s + 8-s − 0.468·9-s + 3.74·10-s + 2.51·11-s − 1.59·12-s − 6.22·13-s − 14-s − 5.96·15-s + 16-s − 2.73·17-s − 0.468·18-s − 1.96·19-s + 3.74·20-s + 1.59·21-s + 2.51·22-s − 0.508·23-s − 1.59·24-s + 9.06·25-s − 6.22·26-s + 5.51·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.918·3-s + 0.5·4-s + 1.67·5-s − 0.649·6-s − 0.377·7-s + 0.353·8-s − 0.156·9-s + 1.18·10-s + 0.759·11-s − 0.459·12-s − 1.72·13-s − 0.267·14-s − 1.54·15-s + 0.250·16-s − 0.664·17-s − 0.110·18-s − 0.450·19-s + 0.838·20-s + 0.347·21-s + 0.536·22-s − 0.106·23-s − 0.324·24-s + 1.81·25-s − 1.22·26-s + 1.06·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 1.59T + 3T^{2} \) |
| 5 | \( 1 - 3.74T + 5T^{2} \) |
| 11 | \( 1 - 2.51T + 11T^{2} \) |
| 13 | \( 1 + 6.22T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 + 1.96T + 19T^{2} \) |
| 23 | \( 1 + 0.508T + 23T^{2} \) |
| 29 | \( 1 + 0.258T + 29T^{2} \) |
| 31 | \( 1 + 4.88T + 31T^{2} \) |
| 37 | \( 1 - 0.469T + 37T^{2} \) |
| 41 | \( 1 + 9.05T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 8.69T + 47T^{2} \) |
| 53 | \( 1 - 0.572T + 53T^{2} \) |
| 59 | \( 1 + 8.62T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 1.39T + 67T^{2} \) |
| 71 | \( 1 + 9.82T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 6.14T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 3.21T + 97T^{2} \) |
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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
−7.27648838519341711256841407591, −6.65706319986178578879532214353, −6.25728979517743210950268466634, −5.52185715214108473625677673625, −5.06774365128745406232837760092, −4.34802127948036619558213323490, −3.09743387749655558341673472293, −2.32553108887746632524351007834, −1.57211115348708598016323971345, 0,
1.57211115348708598016323971345, 2.32553108887746632524351007834, 3.09743387749655558341673472293, 4.34802127948036619558213323490, 5.06774365128745406232837760092, 5.52185715214108473625677673625, 6.25728979517743210950268466634, 6.65706319986178578879532214353, 7.27648838519341711256841407591
