| L(s) = 1 | − 10.3·2-s + 27.9·3-s + 75.8·4-s − 25·5-s − 290.·6-s + 49·7-s − 455.·8-s + 537.·9-s + 259.·10-s − 132.·11-s + 2.12e3·12-s + 802.·13-s − 508.·14-s − 698.·15-s + 2.30e3·16-s + 763.·17-s − 5.58e3·18-s − 61.2·19-s − 1.89e3·20-s + 1.36e3·21-s + 1.37e3·22-s + 3.58e3·23-s − 1.27e4·24-s + 625·25-s − 8.33e3·26-s + 8.23e3·27-s + 3.71e3·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 1.79·3-s + 2.37·4-s − 0.447·5-s − 3.29·6-s + 0.377·7-s − 2.51·8-s + 2.21·9-s + 0.821·10-s − 0.330·11-s + 4.25·12-s + 1.31·13-s − 0.693·14-s − 0.801·15-s + 2.25·16-s + 0.640·17-s − 4.06·18-s − 0.0389·19-s − 1.06·20-s + 0.677·21-s + 0.606·22-s + 1.41·23-s − 4.51·24-s + 0.200·25-s − 2.41·26-s + 2.17·27-s + 0.896·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.251279783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.251279783\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 7 | \( 1 - 49T \) |
| good | 2 | \( 1 + 10.3T + 32T^{2} \) |
| 3 | \( 1 - 27.9T + 243T^{2} \) |
| 11 | \( 1 + 132.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 802.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 763.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 61.2T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.58e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 384.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.37e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.21e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.52e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.36e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.77e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.79e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.56e4T + 8.58e9T^{2} \) |
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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
−15.57887555508836072603751230651, −14.81768849348012148162454530224, −13.21079023549181287334414907098, −11.28966154749385903590000620552, −9.999171450478228112567988881366, −8.797476269090148059454076248012, −8.196252933971707704212009836238, −7.14063555233709744784124230755, −3.22037114632486168490624076771, −1.50248395223322097181713024545,
1.50248395223322097181713024545, 3.22037114632486168490624076771, 7.14063555233709744784124230755, 8.196252933971707704212009836238, 8.797476269090148059454076248012, 9.999171450478228112567988881366, 11.28966154749385903590000620552, 13.21079023549181287334414907098, 14.81768849348012148162454530224, 15.57887555508836072603751230651
