| L(s) = 1 | − 0.605·2-s − 1.63·4-s − 5-s + 3.25·7-s + 2.20·8-s + 0.605·10-s + 0.759·11-s − 0.806·13-s − 1.97·14-s + 1.93·16-s − 5.43·17-s + 0.0290·19-s + 1.63·20-s − 0.460·22-s + 23-s + 25-s + 0.488·26-s − 5.31·28-s − 1.55·29-s + 4.63·31-s − 5.57·32-s + 3.29·34-s − 3.25·35-s + 5.52·37-s − 0.0176·38-s − 2.20·40-s + 5.76·41-s + ⋯ |
| L(s) = 1 | − 0.428·2-s − 0.816·4-s − 0.447·5-s + 1.22·7-s + 0.778·8-s + 0.191·10-s + 0.228·11-s − 0.223·13-s − 0.526·14-s + 0.483·16-s − 1.31·17-s + 0.00666·19-s + 0.365·20-s − 0.0980·22-s + 0.208·23-s + 0.200·25-s + 0.0958·26-s − 1.00·28-s − 0.288·29-s + 0.832·31-s − 0.985·32-s + 0.565·34-s − 0.549·35-s + 0.908·37-s − 0.00285·38-s − 0.347·40-s + 0.899·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.058817710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.058817710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 0.605T + 2T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 - 0.759T + 11T^{2} \) |
| 13 | \( 1 + 0.806T + 13T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 19 | \( 1 - 0.0290T + 19T^{2} \) |
| 29 | \( 1 + 1.55T + 29T^{2} \) |
| 31 | \( 1 - 4.63T + 31T^{2} \) |
| 37 | \( 1 - 5.52T + 37T^{2} \) |
| 41 | \( 1 - 5.76T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 0.299T + 47T^{2} \) |
| 53 | \( 1 + 0.633T + 53T^{2} \) |
| 59 | \( 1 - 5.27T + 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 - 9.24T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 4.80T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 4.30T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 2.40T + 97T^{2} \) |
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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
−9.740917097625371268106644513005, −9.047459377371459927941985350502, −8.258104679038711509862334225557, −7.75975564127036151409271740510, −6.74308982781265614228255488828, −5.42570612019230271050450142687, −4.56110108920229507710171036023, −4.00352462415893642645325960961, −2.31038135078833385316846604135, −0.883173085791483898515254904439,
0.883173085791483898515254904439, 2.31038135078833385316846604135, 4.00352462415893642645325960961, 4.56110108920229507710171036023, 5.42570612019230271050450142687, 6.74308982781265614228255488828, 7.75975564127036151409271740510, 8.258104679038711509862334225557, 9.047459377371459927941985350502, 9.740917097625371268106644513005
