| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 24.6·5-s + 36·6-s + 62.9·7-s − 64·8-s + 81·9-s + 98.6·10-s − 614.·11-s − 144·12-s + 1.12e3·13-s − 251.·14-s + 221.·15-s + 256·16-s + 1.42e3·17-s − 324·18-s − 326.·19-s − 394.·20-s − 566.·21-s + 2.45e3·22-s − 529·23-s + 576·24-s − 2.51e3·25-s − 4.49e3·26-s − 729·27-s + 1.00e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.441·5-s + 0.408·6-s + 0.485·7-s − 0.353·8-s + 0.333·9-s + 0.311·10-s − 1.53·11-s − 0.288·12-s + 1.84·13-s − 0.343·14-s + 0.254·15-s + 0.250·16-s + 1.19·17-s − 0.235·18-s − 0.207·19-s − 0.220·20-s − 0.280·21-s + 1.08·22-s − 0.208·23-s + 0.204·24-s − 0.805·25-s − 1.30·26-s − 0.192·27-s + 0.242·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 + 24.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 62.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 614.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.12e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.42e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 326.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 2.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.27e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.11e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.97e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 920.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.63e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.20e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.76e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.62e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.20e5T + 8.58e9T^{2} \) |
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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
−11.41847014262638724504458809039, −10.82973180377968657785070683695, −9.849459561178718636261274590568, −8.297163343985356056297369460518, −7.78756214860388884750481127580, −6.28107838976836620721893726125, −5.17280270652773787351791871072, −3.44978283741529996837017868636, −1.51213889783129456407477385118, 0,
1.51213889783129456407477385118, 3.44978283741529996837017868636, 5.17280270652773787351791871072, 6.28107838976836620721893726125, 7.78756214860388884750481127580, 8.297163343985356056297369460518, 9.849459561178718636261274590568, 10.82973180377968657785070683695, 11.41847014262638724504458809039
