| L(s) = 1 | − 1.37·2-s − 0.105·4-s + 1.48·5-s + 4.92·7-s + 2.89·8-s − 2.04·10-s + 4.78·11-s + 0.524·13-s − 6.78·14-s − 3.77·16-s + 2.51·17-s + 6.02·19-s − 0.156·20-s − 6.58·22-s + 23-s − 2.80·25-s − 0.722·26-s − 0.519·28-s + 29-s − 4.86·31-s − 0.596·32-s − 3.46·34-s + 7.30·35-s − 1.81·37-s − 8.29·38-s + 4.29·40-s + 6.88·41-s + ⋯ |
| L(s) = 1 | − 0.973·2-s − 0.0527·4-s + 0.663·5-s + 1.86·7-s + 1.02·8-s − 0.645·10-s + 1.44·11-s + 0.145·13-s − 1.81·14-s − 0.944·16-s + 0.610·17-s + 1.38·19-s − 0.0349·20-s − 1.40·22-s + 0.208·23-s − 0.560·25-s − 0.141·26-s − 0.0982·28-s + 0.185·29-s − 0.872·31-s − 0.105·32-s − 0.594·34-s + 1.23·35-s − 0.298·37-s − 1.34·38-s + 0.679·40-s + 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.080455507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.080455507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.37T + 2T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 - 4.78T + 11T^{2} \) |
| 13 | \( 1 - 0.524T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 19 | \( 1 - 6.02T + 19T^{2} \) |
| 31 | \( 1 + 4.86T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 - 6.88T + 41T^{2} \) |
| 43 | \( 1 - 5.44T + 43T^{2} \) |
| 47 | \( 1 - 8.73T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 - 8.40T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 8.80T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 5.81T + 79T^{2} \) |
| 83 | \( 1 + 7.77T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 8.12T + 97T^{2} \) |
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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
−8.232907961399326560883102324509, −7.44652306511221852837585707420, −7.09075968578837026366455250010, −5.73276860185283587198821849662, −5.39045898625422358373096087786, −4.39413793398283651917433526675, −3.85312001064465143826675831717, −2.36519042751161484020830036583, −1.37337487048175230261794793739, −1.12183582808623980891327138323,
1.12183582808623980891327138323, 1.37337487048175230261794793739, 2.36519042751161484020830036583, 3.85312001064465143826675831717, 4.39413793398283651917433526675, 5.39045898625422358373096087786, 5.73276860185283587198821849662, 7.09075968578837026366455250010, 7.44652306511221852837585707420, 8.232907961399326560883102324509
