| L(s) = 1 | + 1.46·2-s − 2.86·3-s + 0.139·4-s + 3.32·5-s − 4.18·6-s − 2.72·8-s + 5.18·9-s + 4.86·10-s + 1.53·11-s − 0.398·12-s + 3.32·13-s − 9.50·15-s − 4.25·16-s − 5.25·17-s + 7.58·18-s − 19-s + 0.462·20-s + 2.24·22-s + 3.60·23-s + 7.78·24-s + 6.04·25-s + 4.86·26-s − 6.24·27-s + 8.83·29-s − 13.9·30-s + 8.18·31-s − 0.786·32-s + ⋯ |
| L(s) = 1 | + 1.03·2-s − 1.65·3-s + 0.0695·4-s + 1.48·5-s − 1.70·6-s − 0.962·8-s + 1.72·9-s + 1.53·10-s + 0.463·11-s − 0.114·12-s + 0.921·13-s − 2.45·15-s − 1.06·16-s − 1.27·17-s + 1.78·18-s − 0.229·19-s + 0.103·20-s + 0.479·22-s + 0.751·23-s + 1.58·24-s + 1.20·25-s + 0.953·26-s − 1.20·27-s + 1.63·29-s − 2.53·30-s + 1.46·31-s − 0.138·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.835769506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.835769506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 3 | \( 1 + 2.86T + 3T^{2} \) |
| 5 | \( 1 - 3.32T + 5T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 - 3.32T + 13T^{2} \) |
| 17 | \( 1 + 5.25T + 17T^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 - 8.83T + 29T^{2} \) |
| 31 | \( 1 - 8.18T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 12.7T + 41T^{2} \) |
| 43 | \( 1 + 6.64T + 43T^{2} \) |
| 47 | \( 1 - 2.11T + 47T^{2} \) |
| 53 | \( 1 + 0.0643T + 53T^{2} \) |
| 59 | \( 1 + 3.04T + 59T^{2} \) |
| 61 | \( 1 - 1.47T + 61T^{2} \) |
| 67 | \( 1 + 2.33T + 67T^{2} \) |
| 71 | \( 1 + 0.526T + 71T^{2} \) |
| 73 | \( 1 - 0.989T + 73T^{2} \) |
| 79 | \( 1 - 0.796T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 5.01T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−10.28756862439806615983782926010, −9.419193221023425129319024361146, −8.609748496589107946999058196601, −6.68173913454549982499838671414, −6.32510792201401614461429516252, −5.78838727157658911647441803205, −4.85680420899505502383213324538, −4.29109301446686981210878307288, −2.65022006294620398753327706982, −1.06625531540110886793134680305,
1.06625531540110886793134680305, 2.65022006294620398753327706982, 4.29109301446686981210878307288, 4.85680420899505502383213324538, 5.78838727157658911647441803205, 6.32510792201401614461429516252, 6.68173913454549982499838671414, 8.609748496589107946999058196601, 9.419193221023425129319024361146, 10.28756862439806615983782926010
