| L(s) = 1 | + 2.59·2-s − 2.89·3-s + 4.72·4-s + 1.85·5-s − 7.52·6-s + 7.08·8-s + 5.40·9-s + 4.82·10-s − 3.51·11-s − 13.7·12-s + 3.68·13-s − 5.38·15-s + 8.91·16-s + 4.85·17-s + 14.0·18-s + 19-s + 8.78·20-s − 9.11·22-s + 1.71·23-s − 20.5·24-s − 1.54·25-s + 9.56·26-s − 6.97·27-s + 3.20·29-s − 13.9·30-s + 0.627·31-s + 8.95·32-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 1.67·3-s + 2.36·4-s + 0.830·5-s − 3.07·6-s + 2.50·8-s + 1.80·9-s + 1.52·10-s − 1.05·11-s − 3.95·12-s + 1.02·13-s − 1.39·15-s + 2.22·16-s + 1.17·17-s + 3.30·18-s + 0.229·19-s + 1.96·20-s − 1.94·22-s + 0.358·23-s − 4.19·24-s − 0.309·25-s + 1.87·26-s − 1.34·27-s + 0.594·29-s − 2.55·30-s + 0.112·31-s + 1.58·32-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
| \(L(1)\) |
\(\approx\) |
\(3.373415492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.373415492\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.59T + 2T^{2} \) |
| 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 - 1.85T + 5T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 - 3.68T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 23 | \( 1 - 1.71T + 23T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 - 0.627T + 31T^{2} \) |
| 37 | \( 1 - 1.62T + 37T^{2} \) |
| 41 | \( 1 + 5.18T + 41T^{2} \) |
| 43 | \( 1 - 6.18T + 43T^{2} \) |
| 47 | \( 1 + 5.43T + 47T^{2} \) |
| 53 | \( 1 - 5.81T + 53T^{2} \) |
| 59 | \( 1 + 4.88T + 59T^{2} \) |
| 61 | \( 1 + 7.36T + 61T^{2} \) |
| 67 | \( 1 + 0.0879T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 - 3.01T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 4.03T + 83T^{2} \) |
| 89 | \( 1 + 1.32T + 89T^{2} \) |
| 97 | \( 1 + 4.69T + 97T^{2} \) |
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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
−10.59399967228385894602110503313, −9.765633465325937844661814405773, −7.945344268271226385505911382341, −6.86990824375657779212425816217, −6.16550060268719941748144294327, −5.51729905298040977737023772190, −5.17377591440523565942354449328, −4.09081617577878455966272008451, −2.86841241801527650304363552166, −1.40646395771418083304972712600,
1.40646395771418083304972712600, 2.86841241801527650304363552166, 4.09081617577878455966272008451, 5.17377591440523565942354449328, 5.51729905298040977737023772190, 6.16550060268719941748144294327, 6.86990824375657779212425816217, 7.945344268271226385505911382341, 9.765633465325937844661814405773, 10.59399967228385894602110503313
