| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 16·5-s − 6·6-s − 28·7-s + 8·8-s + 9·9-s + 32·10-s − 34·11-s − 12·12-s − 56·14-s − 48·15-s + 16·16-s + 138·17-s + 18·18-s − 108·19-s + 64·20-s + 84·21-s − 68·22-s − 52·23-s − 24·24-s + 131·25-s − 27·27-s − 112·28-s − 190·29-s − 96·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.43·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.01·10-s − 0.931·11-s − 0.288·12-s − 1.06·14-s − 0.826·15-s + 1/4·16-s + 1.96·17-s + 0.235·18-s − 1.30·19-s + 0.715·20-s + 0.872·21-s − 0.658·22-s − 0.471·23-s − 0.204·24-s + 1.04·25-s − 0.192·27-s − 0.755·28-s − 1.21·29-s − 0.584·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 138 T + p^{3} T^{2} \) |
| 19 | \( 1 + 108 T + p^{3} T^{2} \) |
| 23 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 - 176 T + p^{3} T^{2} \) |
| 37 | \( 1 + 342 T + p^{3} T^{2} \) |
| 41 | \( 1 + 240 T + p^{3} T^{2} \) |
| 43 | \( 1 + 140 T + p^{3} T^{2} \) |
| 47 | \( 1 + 454 T + p^{3} T^{2} \) |
| 53 | \( 1 - 198 T + p^{3} T^{2} \) |
| 59 | \( 1 - 154 T + p^{3} T^{2} \) |
| 61 | \( 1 - 34 T + p^{3} T^{2} \) |
| 67 | \( 1 - 656 T + p^{3} T^{2} \) |
| 71 | \( 1 + 550 T + p^{3} T^{2} \) |
| 73 | \( 1 + 614 T + p^{3} T^{2} \) |
| 79 | \( 1 - 8 T + p^{3} T^{2} \) |
| 83 | \( 1 + 762 T + p^{3} T^{2} \) |
| 89 | \( 1 - 444 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1022 T + p^{3} T^{2} \) |
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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
−9.582827937702391576103065246347, −8.287493151150060168293832577700, −7.05545538704194162890816979781, −6.32640914284916798230720684255, −5.70816053756138740046661173995, −5.14068526838945189756835408080, −3.68176029589737001381461525591, −2.77667777494192000689085499077, −1.65640861255921842921377102476, 0,
1.65640861255921842921377102476, 2.77667777494192000689085499077, 3.68176029589737001381461525591, 5.14068526838945189756835408080, 5.70816053756138740046661173995, 6.32640914284916798230720684255, 7.05545538704194162890816979781, 8.287493151150060168293832577700, 9.582827937702391576103065246347
