| L(s) = 1 | + 3-s + 3.91·5-s − 1.60·7-s + 9-s + 5.85·11-s + 0.924·13-s + 3.91·15-s + 6.27·17-s + 2.72·19-s − 1.60·21-s − 0.550·23-s + 10.3·25-s + 27-s + 0.556·29-s − 9.36·31-s + 5.85·33-s − 6.29·35-s + 6.50·37-s + 0.924·39-s + 2.46·41-s + 2.43·43-s + 3.91·45-s − 4.96·47-s − 4.40·49-s + 6.27·51-s − 3.80·53-s + 22.8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.74·5-s − 0.608·7-s + 0.333·9-s + 1.76·11-s + 0.256·13-s + 1.01·15-s + 1.52·17-s + 0.625·19-s − 0.351·21-s − 0.114·23-s + 2.06·25-s + 0.192·27-s + 0.103·29-s − 1.68·31-s + 1.01·33-s − 1.06·35-s + 1.06·37-s + 0.147·39-s + 0.384·41-s + 0.371·43-s + 0.583·45-s − 0.724·47-s − 0.629·49-s + 0.878·51-s − 0.523·53-s + 3.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.483682492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.483682492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 3.91T + 5T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 - 5.85T + 11T^{2} \) |
| 13 | \( 1 - 0.924T + 13T^{2} \) |
| 17 | \( 1 - 6.27T + 17T^{2} \) |
| 19 | \( 1 - 2.72T + 19T^{2} \) |
| 23 | \( 1 + 0.550T + 23T^{2} \) |
| 29 | \( 1 - 0.556T + 29T^{2} \) |
| 31 | \( 1 + 9.36T + 31T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 + 4.96T + 47T^{2} \) |
| 53 | \( 1 + 3.80T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 - 2.78T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 7.96T + 89T^{2} \) |
| 97 | \( 1 + 4.68T + 97T^{2} \) |
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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
−7.81351189565473335759246743887, −7.00245781773039814817578322043, −6.40231762133252127228080513759, −5.83617718672071094872571593861, −5.26196488057701636376061706015, −4.11287275562652280827974748590, −3.40082094692878147631990579805, −2.70475034179329602102248464404, −1.62332719205631176961030591016, −1.19206334705591955492771975584,
1.19206334705591955492771975584, 1.62332719205631176961030591016, 2.70475034179329602102248464404, 3.40082094692878147631990579805, 4.11287275562652280827974748590, 5.26196488057701636376061706015, 5.83617718672071094872571593861, 6.40231762133252127228080513759, 7.00245781773039814817578322043, 7.81351189565473335759246743887
