| L(s) = 1 | + 5-s + 5.24·7-s − 5.14·11-s − 2.10·13-s − 5.14·17-s − 7.24·19-s + 6.24·23-s + 25-s − 4.24·29-s + 3.14·31-s + 5.24·35-s + 1.24·37-s − 2.20·41-s + 6.03·43-s − 10.2·47-s + 20.5·49-s + 4·53-s − 5.14·55-s + 6.49·59-s + 1.24·61-s − 2.10·65-s − 5.03·67-s − 1.79·71-s − 11.7·73-s − 26.9·77-s − 6.10·79-s − 14.7·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.98·7-s − 1.55·11-s − 0.583·13-s − 1.24·17-s − 1.66·19-s + 1.30·23-s + 0.200·25-s − 0.788·29-s + 0.564·31-s + 0.886·35-s + 0.205·37-s − 0.344·41-s + 0.921·43-s − 1.49·47-s + 2.93·49-s + 0.549·53-s − 0.693·55-s + 0.845·59-s + 0.159·61-s − 0.260·65-s − 0.615·67-s − 0.212·71-s − 1.37·73-s − 3.07·77-s − 0.686·79-s − 1.61·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 5.24T + 7T^{2} \) |
| 11 | \( 1 + 5.14T + 11T^{2} \) |
| 13 | \( 1 + 2.10T + 13T^{2} \) |
| 17 | \( 1 + 5.14T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 3.14T + 31T^{2} \) |
| 37 | \( 1 - 1.24T + 37T^{2} \) |
| 41 | \( 1 + 2.20T + 41T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 6.49T + 59T^{2} \) |
| 61 | \( 1 - 1.24T + 61T^{2} \) |
| 67 | \( 1 + 5.03T + 67T^{2} \) |
| 71 | \( 1 + 1.79T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 6.10T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 8.28T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−7.44485083194495604616792232369, −6.91318925599539860219257522197, −5.89127414434774327244105289129, −5.20215086090895612360784830076, −4.71507973339579690256163257871, −4.20565767449225558553004406154, −2.67403785059451716865821357843, −2.27161432125032460950637285316, −1.44211316047614096288359826378, 0,
1.44211316047614096288359826378, 2.27161432125032460950637285316, 2.67403785059451716865821357843, 4.20565767449225558553004406154, 4.71507973339579690256163257871, 5.20215086090895612360784830076, 5.89127414434774327244105289129, 6.91318925599539860219257522197, 7.44485083194495604616792232369
