| L(s) = 1 | + 2-s − 3-s + 4-s + 1.83·5-s − 6-s − 1.21·7-s + 8-s + 9-s + 1.83·10-s + 11-s − 12-s − 4.62·13-s − 1.21·14-s − 1.83·15-s + 16-s + 17-s + 18-s − 1.62·19-s + 1.83·20-s + 1.21·21-s + 22-s − 3.62·23-s − 24-s − 1.62·25-s − 4.62·26-s − 27-s − 1.21·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.822·5-s − 0.408·6-s − 0.460·7-s + 0.353·8-s + 0.333·9-s + 0.581·10-s + 0.301·11-s − 0.288·12-s − 1.28·13-s − 0.325·14-s − 0.474·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.371·19-s + 0.411·20-s + 0.265·21-s + 0.213·22-s − 0.754·23-s − 0.204·24-s − 0.324·25-s − 0.906·26-s − 0.192·27-s − 0.230·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 - 1.83T + 5T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 13 | \( 1 + 4.62T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 + 3.62T + 23T^{2} \) |
| 29 | \( 1 + 8.89T + 29T^{2} \) |
| 31 | \( 1 + 6.21T + 31T^{2} \) |
| 37 | \( 1 + 8.05T + 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 - 2.83T + 43T^{2} \) |
| 47 | \( 1 - 7.95T + 47T^{2} \) |
| 53 | \( 1 - 4.40T + 53T^{2} \) |
| 59 | \( 1 + 7.57T + 59T^{2} \) |
| 67 | \( 1 - 6.21T + 67T^{2} \) |
| 71 | \( 1 + 6.21T + 71T^{2} \) |
| 73 | \( 1 + 2.89T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 - 6.13T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.73178993757805577729691324672, −7.20750477721326401825918189796, −6.41428380425225101450061747214, −5.70358881871356866927879269626, −5.31833251900842588515353794314, −4.30624063203648590166672939610, −3.56497943546119986380325937466, −2.39822626927580260040702055062, −1.70460147229285180361580201030, 0,
1.70460147229285180361580201030, 2.39822626927580260040702055062, 3.56497943546119986380325937466, 4.30624063203648590166672939610, 5.31833251900842588515353794314, 5.70358881871356866927879269626, 6.41428380425225101450061747214, 7.20750477721326401825918189796, 7.73178993757805577729691324672
