| L(s) = 1 | + 2-s + 3-s + 4-s + 3.02·5-s + 6-s − 2.69·7-s + 8-s + 9-s + 3.02·10-s − 2.27·11-s + 12-s + 13-s − 2.69·14-s + 3.02·15-s + 16-s − 6.46·17-s + 18-s − 5.63·19-s + 3.02·20-s − 2.69·21-s − 2.27·22-s − 4.09·23-s + 24-s + 4.15·25-s + 26-s + 27-s − 2.69·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.35·5-s + 0.408·6-s − 1.01·7-s + 0.353·8-s + 0.333·9-s + 0.956·10-s − 0.684·11-s + 0.288·12-s + 0.277·13-s − 0.719·14-s + 0.781·15-s + 0.250·16-s − 1.56·17-s + 0.235·18-s − 1.29·19-s + 0.676·20-s − 0.587·21-s − 0.484·22-s − 0.853·23-s + 0.204·24-s + 0.830·25-s + 0.196·26-s + 0.192·27-s − 0.508·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 3.02T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 + 5.63T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 + 8.69T + 29T^{2} \) |
| 31 | \( 1 - 0.130T + 31T^{2} \) |
| 37 | \( 1 + 5.91T + 37T^{2} \) |
| 41 | \( 1 + 3.27T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 1.40T + 59T^{2} \) |
| 61 | \( 1 - 2.06T + 61T^{2} \) |
| 67 | \( 1 - 3.30T + 67T^{2} \) |
| 71 | \( 1 - 3.12T + 71T^{2} \) |
| 73 | \( 1 + 6.65T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 0.455T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−7.28020993911609218958961742638, −6.52316350917192738627663768043, −6.19065218476742800876935407841, −5.48698227484323253458939979083, −4.63161792894548487705364561346, −3.84491956652420808657764551168, −3.08039983034068428438875114444, −2.10377324235724926974740638344, −1.96576134567909217468434000166, 0,
1.96576134567909217468434000166, 2.10377324235724926974740638344, 3.08039983034068428438875114444, 3.84491956652420808657764551168, 4.63161792894548487705364561346, 5.48698227484323253458939979083, 6.19065218476742800876935407841, 6.52316350917192738627663768043, 7.28020993911609218958961742638
