| L(s) = 1 | + 2·3-s − 5-s − 11·7-s − 23·9-s + 18·11-s + 82·13-s − 2·15-s − 6·17-s − 25·19-s − 22·21-s + 58·23-s − 124·25-s − 100·27-s − 180·29-s + 31·31-s + 36·33-s + 11·35-s + 146·37-s + 164·39-s + 47·41-s + 12·43-s + 23·45-s − 136·47-s − 222·49-s − 12·51-s + 232·53-s − 18·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 0.0894·5-s − 0.593·7-s − 0.851·9-s + 0.493·11-s + 1.74·13-s − 0.0344·15-s − 0.0856·17-s − 0.301·19-s − 0.228·21-s + 0.525·23-s − 0.991·25-s − 0.712·27-s − 1.15·29-s + 0.179·31-s + 0.189·33-s + 0.0531·35-s + 0.648·37-s + 0.673·39-s + 0.179·41-s + 0.0425·43-s + 0.0761·45-s − 0.422·47-s − 0.647·49-s − 0.0329·51-s + 0.601·53-s − 0.0441·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 + 11 T + p^{3} T^{2} \) |
| 11 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 82 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 + 25 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + 180 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 47 T + p^{3} T^{2} \) |
| 43 | \( 1 - 12 T + p^{3} T^{2} \) |
| 47 | \( 1 + 136 T + p^{3} T^{2} \) |
| 53 | \( 1 - 232 T + p^{3} T^{2} \) |
| 59 | \( 1 + 715 T + p^{3} T^{2} \) |
| 61 | \( 1 - 518 T + p^{3} T^{2} \) |
| 67 | \( 1 - 436 T + p^{3} T^{2} \) |
| 71 | \( 1 - 387 T + p^{3} T^{2} \) |
| 73 | \( 1 - 678 T + p^{3} T^{2} \) |
| 79 | \( 1 - 660 T + p^{3} T^{2} \) |
| 83 | \( 1 - 382 T + p^{3} T^{2} \) |
| 89 | \( 1 + 800 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1631 T + p^{3} T^{2} \) |
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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
−8.416089344174517439848577345648, −7.86625913133952878459368045469, −6.67617482572258890607920273580, −6.13230859509782357222004674483, −5.35245911396319329420701995310, −3.94838939184145128024380389637, −3.52819769430534864452598647562, −2.46876720775046336767675192298, −1.28474705597995180875405758329, 0,
1.28474705597995180875405758329, 2.46876720775046336767675192298, 3.52819769430534864452598647562, 3.94838939184145128024380389637, 5.35245911396319329420701995310, 6.13230859509782357222004674483, 6.67617482572258890607920273580, 7.86625913133952878459368045469, 8.416089344174517439848577345648
