| L(s) = 1 | − 2.61·2-s + 2.61·3-s + 4.85·4-s − 2.23·5-s − 6.85·6-s − 7.47·8-s + 3.85·9-s + 5.85·10-s − 5.61·11-s + 12.7·12-s − 13-s − 5.85·15-s + 9.85·16-s + 4.85·17-s − 10.0·18-s + 19-s − 10.8·20-s + 14.7·22-s − 3·23-s − 19.5·24-s + 2.61·26-s + 2.23·27-s − 5.61·29-s + 15.3·30-s + 0.854·31-s − 10.8·32-s − 14.7·33-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 1.51·3-s + 2.42·4-s − 0.999·5-s − 2.79·6-s − 2.64·8-s + 1.28·9-s + 1.85·10-s − 1.69·11-s + 3.66·12-s − 0.277·13-s − 1.51·15-s + 2.46·16-s + 1.17·17-s − 2.37·18-s + 0.229·19-s − 2.42·20-s + 3.13·22-s − 0.625·23-s − 3.99·24-s + 0.513·26-s + 0.430·27-s − 1.04·29-s + 2.79·30-s + 0.153·31-s − 1.91·32-s − 2.56·33-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 11 | \( 1 + 5.61T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 5.61T + 29T^{2} \) |
| 31 | \( 1 - 0.854T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 0.381T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 + 0.708T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 - 6.56T + 67T^{2} \) |
| 71 | \( 1 - 7.47T + 71T^{2} \) |
| 73 | \( 1 + 4.14T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 7.85T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−9.651949754751308661009286418549, −8.546380789600828505643287060300, −8.077397576478401044647518747403, −7.68968138680611923618704581109, −7.01714245933988477690687945189, −5.40171319723633680018681200062, −3.62922633705583399334905222360, −2.84310425131241536948304330437, −1.83012812850839445151614616167, 0,
1.83012812850839445151614616167, 2.84310425131241536948304330437, 3.62922633705583399334905222360, 5.40171319723633680018681200062, 7.01714245933988477690687945189, 7.68968138680611923618704581109, 8.077397576478401044647518747403, 8.546380789600828505643287060300, 9.651949754751308661009286418549
