| L(s) = 1 | − 0.381·5-s + 4.23·7-s − 2.23·13-s − 1.85·17-s − 0.381·19-s − 6.23·23-s − 4.85·25-s + 0.472·29-s + 8.61·31-s − 1.61·35-s + 3.76·37-s − 5·41-s − 11.4·43-s − 0.145·47-s + 10.9·49-s + 9.56·53-s − 14.5·59-s + 6.85·61-s + 0.854·65-s − 4.56·67-s − 12.5·71-s − 3.70·73-s − 14.2·79-s − 1.94·83-s + 0.708·85-s − 3.47·89-s − 9.47·91-s + ⋯ |
| L(s) = 1 | − 0.170·5-s + 1.60·7-s − 0.620·13-s − 0.449·17-s − 0.0876·19-s − 1.30·23-s − 0.970·25-s + 0.0876·29-s + 1.54·31-s − 0.273·35-s + 0.618·37-s − 0.780·41-s − 1.74·43-s − 0.0212·47-s + 1.56·49-s + 1.31·53-s − 1.89·59-s + 0.877·61-s + 0.105·65-s − 0.557·67-s − 1.49·71-s − 0.434·73-s − 1.60·79-s − 0.213·83-s + 0.0768·85-s − 0.368·89-s − 0.992·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 0.381T + 5T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 + 0.381T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 - 8.61T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 0.145T + 47T^{2} \) |
| 53 | \( 1 - 9.56T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 3.70T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 1.94T + 83T^{2} \) |
| 89 | \( 1 + 3.47T + 89T^{2} \) |
| 97 | \( 1 + 6.85T + 97T^{2} \) |
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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
−7.55234756349822344192487515269, −6.82515368521836295161232497533, −5.97109598148945062108651742999, −5.28973504756046097630218544330, −4.47142109113364068717389512294, −4.20254175871053739826463900243, −2.96502383778221538621355275505, −2.07906693453328790050939110432, −1.41387662592598457157868499456, 0,
1.41387662592598457157868499456, 2.07906693453328790050939110432, 2.96502383778221538621355275505, 4.20254175871053739826463900243, 4.47142109113364068717389512294, 5.28973504756046097630218544330, 5.97109598148945062108651742999, 6.82515368521836295161232497533, 7.55234756349822344192487515269
