L(s) = 1 | − 4.53·2-s − 11.4·4-s + 25·5-s − 49·7-s + 196.·8-s − 113.·10-s − 90.5·11-s − 74.8·13-s + 222.·14-s − 525.·16-s − 1.03e3·17-s − 31.6·19-s − 286.·20-s + 410.·22-s + 3.85e3·23-s + 625·25-s + 339.·26-s + 561.·28-s + 866.·29-s + 3.52e3·31-s − 3.92e3·32-s + 4.67e3·34-s − 1.22e3·35-s − 9.53e3·37-s + 143.·38-s + 4.92e3·40-s + 1.45e4·41-s + ⋯ |
L(s) = 1 | − 0.800·2-s − 0.358·4-s + 0.447·5-s − 0.377·7-s + 1.08·8-s − 0.358·10-s − 0.225·11-s − 0.122·13-s + 0.302·14-s − 0.513·16-s − 0.866·17-s − 0.0201·19-s − 0.160·20-s + 0.180·22-s + 1.52·23-s + 0.200·25-s + 0.0983·26-s + 0.135·28-s + 0.191·29-s + 0.659·31-s − 0.677·32-s + 0.693·34-s − 0.169·35-s − 1.14·37-s + 0.0161·38-s + 0.486·40-s + 1.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 + 49T \) |
good | 2 | \( 1 + 4.53T + 32T^{2} \) |
| 11 | \( 1 + 90.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 74.8T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 31.6T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.85e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 866.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.53e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.45e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.84e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.96e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.92e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.51e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.18e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.03e3T + 8.58e9T^{2} \) |
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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
−10.30070572094034656581720707076, −9.229467752114237855640760901308, −8.792427168583566349439060444848, −7.58311375807142811795467682041, −6.64723424087362840971323701027, −5.32597050109335014363708857242, −4.28836160925783180451674140082, −2.72616703758039750280551055310, −1.27176536806453243952899125631, 0,
1.27176536806453243952899125631, 2.72616703758039750280551055310, 4.28836160925783180451674140082, 5.32597050109335014363708857242, 6.64723424087362840971323701027, 7.58311375807142811795467682041, 8.792427168583566349439060444848, 9.229467752114237855640760901308, 10.30070572094034656581720707076