| L(s) = 1 | + 1.09·2-s + 2.77·3-s − 0.800·4-s + 1.94·5-s + 3.03·6-s − 3.06·8-s + 4.68·9-s + 2.12·10-s + 3.67·11-s − 2.21·12-s − 2.48·13-s + 5.38·15-s − 1.75·16-s + 4.50·17-s + 5.13·18-s + 7.12·19-s − 1.55·20-s + 4.02·22-s − 4.99·23-s − 8.50·24-s − 1.23·25-s − 2.72·26-s + 4.67·27-s + 4.78·29-s + 5.89·30-s + 1.30·31-s + 4.20·32-s + ⋯ |
| L(s) = 1 | + 0.774·2-s + 1.60·3-s − 0.400·4-s + 0.868·5-s + 1.23·6-s − 1.08·8-s + 1.56·9-s + 0.672·10-s + 1.10·11-s − 0.640·12-s − 0.689·13-s + 1.38·15-s − 0.439·16-s + 1.09·17-s + 1.20·18-s + 1.63·19-s − 0.347·20-s + 0.857·22-s − 1.04·23-s − 1.73·24-s − 0.246·25-s − 0.533·26-s + 0.899·27-s + 0.888·29-s + 1.07·30-s + 0.233·31-s + 0.743·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6713 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6713 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.083516989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.083516989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 137 | \( 1 + T \) |
| good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 3 | \( 1 - 2.77T + 3T^{2} \) |
| 5 | \( 1 - 1.94T + 5T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 13 | \( 1 + 2.48T + 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 - 4.78T + 29T^{2} \) |
| 31 | \( 1 - 1.30T + 31T^{2} \) |
| 37 | \( 1 - 7.31T + 37T^{2} \) |
| 41 | \( 1 + 6.83T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 8.28T + 47T^{2} \) |
| 53 | \( 1 + 3.20T + 53T^{2} \) |
| 59 | \( 1 + 1.64T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 - 9.53T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 6.85T + 89T^{2} \) |
| 97 | \( 1 + 4.61T + 97T^{2} \) |
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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
−8.120786284748017866723284196702, −7.36316676890076242476004174744, −6.51827021791866351913693255839, −5.73492316381720385652776492277, −5.06052300026943787233031363730, −4.19069811791674688904127004396, −3.50243144952319275637150772621, −2.97845253080987469393249291108, −2.10691827137864366654429777860, −1.13529168172357943036968103193,
1.13529168172357943036968103193, 2.10691827137864366654429777860, 2.97845253080987469393249291108, 3.50243144952319275637150772621, 4.19069811791674688904127004396, 5.06052300026943787233031363730, 5.73492316381720385652776492277, 6.51827021791866351913693255839, 7.36316676890076242476004174744, 8.120786284748017866723284196702
