| L(s) = 1 | + 3.12·3-s − 3.76·5-s + 2.42·7-s + 6.77·9-s + 6.57·11-s + 13-s − 11.7·15-s − 1.11·17-s − 3.43·19-s + 7.57·21-s − 1.69·23-s + 9.16·25-s + 11.7·27-s − 29-s − 0.408·31-s + 20.5·33-s − 9.11·35-s + 11.7·37-s + 3.12·39-s + 3.63·41-s + 8.08·43-s − 25.4·45-s − 5.68·47-s − 1.13·49-s − 3.49·51-s + 1.27·53-s − 24.7·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s − 1.68·5-s + 0.915·7-s + 2.25·9-s + 1.98·11-s + 0.277·13-s − 3.03·15-s − 0.270·17-s − 0.787·19-s + 1.65·21-s − 0.353·23-s + 1.83·25-s + 2.26·27-s − 0.185·29-s − 0.0733·31-s + 3.57·33-s − 1.54·35-s + 1.92·37-s + 0.500·39-s + 0.568·41-s + 1.23·43-s − 3.79·45-s − 0.828·47-s − 0.161·49-s − 0.488·51-s + 0.174·53-s − 3.33·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.920945132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.920945132\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 3.12T + 3T^{2} \) |
| 5 | \( 1 + 3.76T + 5T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 - 6.57T + 11T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 + 3.43T + 19T^{2} \) |
| 23 | \( 1 + 1.69T + 23T^{2} \) |
| 31 | \( 1 + 0.408T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 3.63T + 41T^{2} \) |
| 43 | \( 1 - 8.08T + 43T^{2} \) |
| 47 | \( 1 + 5.68T + 47T^{2} \) |
| 53 | \( 1 - 1.27T + 53T^{2} \) |
| 59 | \( 1 + 7.03T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 + 0.218T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 6.83T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 9.23T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.034408402141774738854604649454, −7.72285802094112091493199092542, −6.96654280981857619985798040620, −6.23882287510837256974849410307, −4.57222363370145541045443801884, −4.18818005562940739097036850790, −3.79047566743502163621006931673, −2.94273467334080868657887300226, −1.90966070490207513431132980905, −1.03410229667270366405955682200,
1.03410229667270366405955682200, 1.90966070490207513431132980905, 2.94273467334080868657887300226, 3.79047566743502163621006931673, 4.18818005562940739097036850790, 4.57222363370145541045443801884, 6.23882287510837256974849410307, 6.96654280981857619985798040620, 7.72285802094112091493199092542, 8.034408402141774738854604649454
