| L(s) = 1 | + 3.15·3-s + 0.719·5-s + 3.90·7-s + 6.98·9-s − 0.402·11-s + 1.41·13-s + 2.27·15-s − 17-s + 5.84·19-s + 12.3·21-s − 2.22·23-s − 4.48·25-s + 12.5·27-s − 2.13·29-s + 4.81·31-s − 1.27·33-s + 2.81·35-s − 10.4·37-s + 4.46·39-s + 1.82·41-s − 3.94·43-s + 5.02·45-s + 8.01·47-s + 8.26·49-s − 3.15·51-s + 3.18·53-s − 0.289·55-s + ⋯ |
| L(s) = 1 | + 1.82·3-s + 0.321·5-s + 1.47·7-s + 2.32·9-s − 0.121·11-s + 0.392·13-s + 0.587·15-s − 0.242·17-s + 1.34·19-s + 2.69·21-s − 0.463·23-s − 0.896·25-s + 2.42·27-s − 0.397·29-s + 0.864·31-s − 0.221·33-s + 0.475·35-s − 1.71·37-s + 0.715·39-s + 0.285·41-s − 0.602·43-s + 0.749·45-s + 1.16·47-s + 1.18·49-s − 0.442·51-s + 0.437·53-s − 0.0390·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.918803576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.918803576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 3.15T + 3T^{2} \) |
| 5 | \( 1 - 0.719T + 5T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 11 | \( 1 + 0.402T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 19 | \( 1 - 5.84T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 + 2.13T + 29T^{2} \) |
| 31 | \( 1 - 4.81T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 43 | \( 1 + 3.94T + 43T^{2} \) |
| 47 | \( 1 - 8.01T + 47T^{2} \) |
| 53 | \( 1 - 3.18T + 53T^{2} \) |
| 61 | \( 1 + 6.79T + 61T^{2} \) |
| 67 | \( 1 + 9.62T + 67T^{2} \) |
| 71 | \( 1 - 5.24T + 71T^{2} \) |
| 73 | \( 1 - 5.73T + 73T^{2} \) |
| 79 | \( 1 - 9.27T + 79T^{2} \) |
| 83 | \( 1 - 5.97T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 2.51T + 97T^{2} \) |
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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
−7.954690359767560009687752916567, −7.49662450751524490206244803041, −6.71349266887992185664387036751, −5.59559772755441392548020049921, −4.91140124920998692363115490701, −4.09380477028004210298864314717, −3.48410521937067927419529996003, −2.57538691089444658103849830091, −1.87626456821506311618888411323, −1.26407046473454395498418813072,
1.26407046473454395498418813072, 1.87626456821506311618888411323, 2.57538691089444658103849830091, 3.48410521937067927419529996003, 4.09380477028004210298864314717, 4.91140124920998692363115490701, 5.59559772755441392548020049921, 6.71349266887992185664387036751, 7.49662450751524490206244803041, 7.954690359767560009687752916567
