| L(s) = 1 | − 8.45·2-s + 39.4·4-s + 58.5·5-s + 96.1·7-s − 63.0·8-s − 495.·10-s − 584.·11-s + 73.5·13-s − 812.·14-s − 729.·16-s − 1.72e3·17-s + 1.75e3·19-s + 2.31e3·20-s + 4.94e3·22-s − 529·23-s + 307.·25-s − 622.·26-s + 3.79e3·28-s − 6.00e3·29-s − 776.·31-s + 8.18e3·32-s + 1.46e4·34-s + 5.63e3·35-s + 1.47e4·37-s − 1.48e4·38-s − 3.69e3·40-s + 4.56e3·41-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.23·4-s + 1.04·5-s + 0.741·7-s − 0.348·8-s − 1.56·10-s − 1.45·11-s + 0.120·13-s − 1.10·14-s − 0.712·16-s − 1.44·17-s + 1.11·19-s + 1.29·20-s + 2.17·22-s − 0.208·23-s + 0.0984·25-s − 0.180·26-s + 0.914·28-s − 1.32·29-s − 0.145·31-s + 1.41·32-s + 2.16·34-s + 0.777·35-s + 1.77·37-s − 1.66·38-s − 0.364·40-s + 0.423·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 8.45T + 32T^{2} \) |
| 5 | \( 1 - 58.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 96.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 584.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 73.5T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.72e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.75e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 6.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 776.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.47e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.56e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.44e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.63e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.59e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.41e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.56e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.29e4T + 8.58e9T^{2} \) |
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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
−10.79903403231145536273094886058, −9.873291486415207035574710684349, −9.170039946367393756762280619235, −8.108195092167331626450754921614, −7.36489359279518636121362041535, −5.96880649121293274077725447754, −4.79389893613117813321014694641, −2.48731597897088906989121684809, −1.55403519700964290308520471939, 0,
1.55403519700964290308520471939, 2.48731597897088906989121684809, 4.79389893613117813321014694641, 5.96880649121293274077725447754, 7.36489359279518636121362041535, 8.108195092167331626450754921614, 9.170039946367393756762280619235, 9.873291486415207035574710684349, 10.79903403231145536273094886058
