| L(s) = 1 | − 1.31·2-s − 3.01·3-s − 0.271·4-s + 4.02·5-s + 3.96·6-s + 2.98·8-s + 6.08·9-s − 5.29·10-s + 1.61·11-s + 0.817·12-s + 5.21·13-s − 12.1·15-s − 3.38·16-s + 2.70·17-s − 8.00·18-s + 19-s − 1.09·20-s − 2.12·22-s − 2.88·23-s − 9.00·24-s + 11.2·25-s − 6.86·26-s − 9.30·27-s − 6.81·29-s + 15.9·30-s + 0.440·31-s − 1.52·32-s + ⋯ |
| L(s) = 1 | − 0.929·2-s − 1.74·3-s − 0.135·4-s + 1.80·5-s + 1.61·6-s + 1.05·8-s + 2.02·9-s − 1.67·10-s + 0.486·11-s + 0.236·12-s + 1.44·13-s − 3.13·15-s − 0.846·16-s + 0.656·17-s − 1.88·18-s + 0.229·19-s − 0.244·20-s − 0.452·22-s − 0.602·23-s − 1.83·24-s + 2.24·25-s − 1.34·26-s − 1.79·27-s − 1.26·29-s + 2.91·30-s + 0.0791·31-s − 0.269·32-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
| \(L(1)\) |
\(\approx\) |
\(0.7802281392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7802281392\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.31T + 2T^{2} \) |
| 3 | \( 1 + 3.01T + 3T^{2} \) |
| 5 | \( 1 - 4.02T + 5T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 - 5.21T + 13T^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 23 | \( 1 + 2.88T + 23T^{2} \) |
| 29 | \( 1 + 6.81T + 29T^{2} \) |
| 31 | \( 1 - 0.440T + 31T^{2} \) |
| 37 | \( 1 + 6.68T + 37T^{2} \) |
| 41 | \( 1 - 4.42T + 41T^{2} \) |
| 43 | \( 1 - 8.88T + 43T^{2} \) |
| 47 | \( 1 - 0.655T + 47T^{2} \) |
| 53 | \( 1 + 4.75T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 3.63T + 71T^{2} \) |
| 73 | \( 1 + 6.49T + 73T^{2} \) |
| 79 | \( 1 - 8.78T + 79T^{2} \) |
| 83 | \( 1 - 0.609T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 6.91T + 97T^{2} \) |
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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
−10.09540143964795977215505913721, −9.456588262906850787467046198203, −8.722897546546473277036467917356, −7.39204818659642661176840359109, −6.42519689207237085617782639428, −5.79628395308982604919025764250, −5.23226447373775647316761283442, −3.99556594507382982122535794133, −1.75253059812865661759487006579, −0.976438133721178463693765127386,
0.976438133721178463693765127386, 1.75253059812865661759487006579, 3.99556594507382982122535794133, 5.23226447373775647316761283442, 5.79628395308982604919025764250, 6.42519689207237085617782639428, 7.39204818659642661176840359109, 8.722897546546473277036467917356, 9.456588262906850787467046198203, 10.09540143964795977215505913721
