| L(s) = 1 | + 2-s − 0.00804·3-s + 4-s + 3.67·5-s − 0.00804·6-s + 7-s + 8-s − 2.99·9-s + 3.67·10-s − 3.60·11-s − 0.00804·12-s − 5.05·13-s + 14-s − 0.0295·15-s + 16-s − 4.25·17-s − 2.99·18-s − 7.20·19-s + 3.67·20-s − 0.00804·21-s − 3.60·22-s − 5.45·23-s − 0.00804·24-s + 8.51·25-s − 5.05·26-s + 0.0482·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.00464·3-s + 0.5·4-s + 1.64·5-s − 0.00328·6-s + 0.377·7-s + 0.353·8-s − 0.999·9-s + 1.16·10-s − 1.08·11-s − 0.00232·12-s − 1.40·13-s + 0.267·14-s − 0.00763·15-s + 0.250·16-s − 1.03·17-s − 0.707·18-s − 1.65·19-s + 0.822·20-s − 0.00175·21-s − 0.768·22-s − 1.13·23-s − 0.00164·24-s + 1.70·25-s − 0.992·26-s + 0.00928·27-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 | \( 1 + 0.00804T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 11 | \( 1 + 3.60T + 11T^{2} \) |
| 13 | \( 1 + 5.05T + 13T^{2} \) |
| 17 | \( 1 + 4.25T + 17T^{2} \) |
| 19 | \( 1 + 7.20T + 19T^{2} \) |
| 23 | \( 1 + 5.45T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 - 8.43T + 31T^{2} \) |
| 37 | \( 1 + 6.72T + 37T^{2} \) |
| 41 | \( 1 + 2.42T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 6.86T + 53T^{2} \) |
| 59 | \( 1 + 3.53T + 59T^{2} \) |
| 61 | \( 1 + 1.04T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 9.30T + 71T^{2} \) |
| 73 | \( 1 + 2.31T + 73T^{2} \) |
| 79 | \( 1 - 2.18T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 3.87T + 89T^{2} \) |
| 97 | \( 1 + 5.64T + 97T^{2} \) |
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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
−7.67551911286199395864290394072, −6.70521592181048366164934736004, −6.14567835242686880967970296735, −5.60676691588159838730675636335, −4.84435073303547078826051090395, −4.40607850787025066873999791385, −2.78174480952981874266071900061, −2.49099764044841224466396242186, −1.83245260363763619703014408093, 0,
1.83245260363763619703014408093, 2.49099764044841224466396242186, 2.78174480952981874266071900061, 4.40607850787025066873999791385, 4.84435073303547078826051090395, 5.60676691588159838730675636335, 6.14567835242686880967970296735, 6.70521592181048366164934736004, 7.67551911286199395864290394072
