| L(s) = 1 | − 6.39e7·3-s − 2.18e11·5-s − 4.92e13·7-s − 1.47e15·9-s − 2.23e17·11-s + 7.57e17·13-s + 1.39e19·15-s − 6.49e18·17-s − 7.92e20·19-s + 3.14e21·21-s + 2.99e22·23-s − 6.84e22·25-s + 4.49e23·27-s + 1.97e24·29-s + 4.54e24·31-s + 1.42e25·33-s + 1.07e25·35-s + 1.28e26·37-s − 4.84e25·39-s − 3.19e26·41-s − 1.35e27·43-s + 3.22e26·45-s − 3.49e27·47-s − 5.30e27·49-s + 4.15e26·51-s + 3.47e28·53-s + 4.88e28·55-s + ⋯ |
| L(s) = 1 | − 0.857·3-s − 0.641·5-s − 0.560·7-s − 0.264·9-s − 1.46·11-s + 0.315·13-s + 0.550·15-s − 0.0323·17-s − 0.630·19-s + 0.480·21-s + 1.01·23-s − 0.588·25-s + 1.08·27-s + 1.46·29-s + 1.12·31-s + 1.25·33-s + 0.359·35-s + 1.70·37-s − 0.270·39-s − 0.782·41-s − 1.50·43-s + 0.169·45-s − 0.898·47-s − 0.686·49-s + 0.0277·51-s + 1.23·53-s + 0.939·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(0.6332395951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6332395951\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 6.39e7T + 5.55e15T^{2} \) |
| 5 | \( 1 + 2.18e11T + 1.16e23T^{2} \) |
| 7 | \( 1 + 4.92e13T + 7.73e27T^{2} \) |
| 11 | \( 1 + 2.23e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 7.57e17T + 5.75e36T^{2} \) |
| 17 | \( 1 + 6.49e18T + 4.02e40T^{2} \) |
| 19 | \( 1 + 7.92e20T + 1.58e42T^{2} \) |
| 23 | \( 1 - 2.99e22T + 8.65e44T^{2} \) |
| 29 | \( 1 - 1.97e24T + 1.81e48T^{2} \) |
| 31 | \( 1 - 4.54e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 1.28e26T + 5.63e51T^{2} \) |
| 41 | \( 1 + 3.19e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 1.35e27T + 8.02e53T^{2} \) |
| 47 | \( 1 + 3.49e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 3.47e28T + 7.96e56T^{2} \) |
| 59 | \( 1 - 1.26e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 3.88e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 1.28e30T + 1.82e60T^{2} \) |
| 71 | \( 1 + 2.52e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 8.98e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 9.32e30T + 4.18e62T^{2} \) |
| 83 | \( 1 - 1.92e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 6.91e31T + 2.13e64T^{2} \) |
| 97 | \( 1 - 8.57e32T + 3.65e65T^{2} \) |
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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
−16.58536095215080620425668641927, −15.38972816538049612403309885469, −13.17541531217090719511557393482, −11.69519008438609794438957379778, −10.37787385145013433546168706059, −8.198090569384290375168761949202, −6.38607836330697214818893296773, −4.86824427702834250115861835213, −2.92070769047758086335416690277, −0.50970554940105003785325212622,
0.50970554940105003785325212622, 2.92070769047758086335416690277, 4.86824427702834250115861835213, 6.38607836330697214818893296773, 8.198090569384290375168761949202, 10.37787385145013433546168706059, 11.69519008438609794438957379778, 13.17541531217090719511557393482, 15.38972816538049612403309885469, 16.58536095215080620425668641927
