| L(s) = 1 | + 2·2-s + 4·4-s − 19·7-s + 8·8-s + 12·11-s + 50·13-s − 38·14-s + 16·16-s − 126·17-s + 29·19-s + 24·22-s + 18·23-s + 100·26-s − 76·28-s + 102·29-s − 265·31-s + 32·32-s − 252·34-s + 65·37-s + 58·38-s + 240·41-s − 367·43-s + 48·44-s + 36·46-s − 72·47-s + 18·49-s + 200·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.02·7-s + 0.353·8-s + 0.328·11-s + 1.06·13-s − 0.725·14-s + 1/4·16-s − 1.79·17-s + 0.350·19-s + 0.232·22-s + 0.163·23-s + 0.754·26-s − 0.512·28-s + 0.653·29-s − 1.53·31-s + 0.176·32-s − 1.27·34-s + 0.288·37-s + 0.247·38-s + 0.914·41-s − 1.30·43-s + 0.164·44-s + 0.115·46-s − 0.223·47-s + 0.0524·49-s + 0.533·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 19 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 + 126 T + p^{3} T^{2} \) |
| 19 | \( 1 - 29 T + p^{3} T^{2} \) |
| 23 | \( 1 - 18 T + p^{3} T^{2} \) |
| 29 | \( 1 - 102 T + p^{3} T^{2} \) |
| 31 | \( 1 + 265 T + p^{3} T^{2} \) |
| 37 | \( 1 - 65 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 + 367 T + p^{3} T^{2} \) |
| 47 | \( 1 + 72 T + p^{3} T^{2} \) |
| 53 | \( 1 + 12 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 102 T + p^{3} T^{2} \) |
| 61 | \( 1 + 103 T + p^{3} T^{2} \) |
| 67 | \( 1 + 52 T + p^{3} T^{2} \) |
| 71 | \( 1 + 582 T + p^{3} T^{2} \) |
| 73 | \( 1 - 65 T + p^{3} T^{2} \) |
| 79 | \( 1 - 173 T + p^{3} T^{2} \) |
| 83 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 89 | \( 1 + 822 T + p^{3} T^{2} \) |
| 97 | \( 1 - 821 T + p^{3} T^{2} \) |
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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
−8.968453493668502909444837042867, −7.967368217544327763565051780878, −6.74512022840607179902426969204, −6.49575135535683957383626631110, −5.52859281968176018960570955983, −4.43438311362850857927847518703, −3.64630945244691976823337288898, −2.78048714289531601692795305070, −1.54029925273395022625184063734, 0,
1.54029925273395022625184063734, 2.78048714289531601692795305070, 3.64630945244691976823337288898, 4.43438311362850857927847518703, 5.52859281968176018960570955983, 6.49575135535683957383626631110, 6.74512022840607179902426969204, 7.967368217544327763565051780878, 8.968453493668502909444837042867
