| L(s) = 1 | + 10.3·2-s + 75.1·4-s − 71.3·5-s − 210.·7-s + 447.·8-s − 738.·10-s − 170.·11-s + 605.·13-s − 2.18e3·14-s + 2.22e3·16-s − 1.33e3·17-s − 1.79e3·19-s − 5.36e3·20-s − 1.76e3·22-s − 529·23-s + 1.96e3·25-s + 6.26e3·26-s − 1.58e4·28-s − 3.00e3·29-s − 9.21e3·31-s + 8.71e3·32-s − 1.38e4·34-s + 1.50e4·35-s + 6.02e3·37-s − 1.85e4·38-s − 3.19e4·40-s − 6.60e3·41-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 2.34·4-s − 1.27·5-s − 1.62·7-s + 2.47·8-s − 2.33·10-s − 0.424·11-s + 0.993·13-s − 2.97·14-s + 2.17·16-s − 1.12·17-s − 1.14·19-s − 2.99·20-s − 0.776·22-s − 0.208·23-s + 0.629·25-s + 1.81·26-s − 3.82·28-s − 0.663·29-s − 1.72·31-s + 1.50·32-s − 2.05·34-s + 2.07·35-s + 0.723·37-s − 2.08·38-s − 3.15·40-s − 0.613·41-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 | \( 1 - 10.3T + 32T^{2} \) |
| 5 | \( 1 + 71.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 210.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 170.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 605.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.33e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.79e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 3.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.60e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.60e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.71e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.16e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.78e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.32e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.26e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.14e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.51e5T + 8.58e9T^{2} \) |
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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
−11.26486470309389150944469914363, −10.67882587356178493954322771232, −8.955950414986499548227077451443, −7.47913046886885416101034936980, −6.58519047015384710689287011933, −5.73802192753691010701485149573, −4.12488055849168636666697130989, −3.72333772401368936468949156898, −2.50259158402337504641777216676, 0,
2.50259158402337504641777216676, 3.72333772401368936468949156898, 4.12488055849168636666697130989, 5.73802192753691010701485149573, 6.58519047015384710689287011933, 7.47913046886885416101034936980, 8.955950414986499548227077451443, 10.67882587356178493954322771232, 11.26486470309389150944469914363
