| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 7·5-s − 6·6-s − 15·7-s + 8·8-s + 9·9-s − 14·10-s − 49·11-s − 12·12-s − 14·13-s − 30·14-s + 21·15-s + 16·16-s − 33·17-s + 18·18-s − 28·20-s + 45·21-s − 98·22-s − 148·23-s − 24·24-s − 76·25-s − 28·26-s − 27·27-s − 60·28-s + 278·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.626·5-s − 0.408·6-s − 0.809·7-s + 0.353·8-s + 1/3·9-s − 0.442·10-s − 1.34·11-s − 0.288·12-s − 0.298·13-s − 0.572·14-s + 0.361·15-s + 1/4·16-s − 0.470·17-s + 0.235·18-s − 0.313·20-s + 0.467·21-s − 0.949·22-s − 1.34·23-s − 0.204·24-s − 0.607·25-s − 0.211·26-s − 0.192·27-s − 0.404·28-s + 1.78·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8403497874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8403497874\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 15 T + p^{3} T^{2} \) |
| 11 | \( 1 + 49 T + p^{3} T^{2} \) |
| 13 | \( 1 + 14 T + p^{3} T^{2} \) |
| 17 | \( 1 + 33 T + p^{3} T^{2} \) |
| 23 | \( 1 + 148 T + p^{3} T^{2} \) |
| 29 | \( 1 - 278 T + p^{3} T^{2} \) |
| 31 | \( 1 + 94 T + p^{3} T^{2} \) |
| 37 | \( 1 + 160 T + p^{3} T^{2} \) |
| 41 | \( 1 + 400 T + p^{3} T^{2} \) |
| 43 | \( 1 - 73 T + p^{3} T^{2} \) |
| 47 | \( 1 - 173 T + p^{3} T^{2} \) |
| 53 | \( 1 + 170 T + p^{3} T^{2} \) |
| 59 | \( 1 - 12 T + p^{3} T^{2} \) |
| 61 | \( 1 - 419 T + p^{3} T^{2} \) |
| 67 | \( 1 + 444 T + p^{3} T^{2} \) |
| 71 | \( 1 - 952 T + p^{3} T^{2} \) |
| 73 | \( 1 + 27 T + p^{3} T^{2} \) |
| 79 | \( 1 - 556 T + p^{3} T^{2} \) |
| 83 | \( 1 + 276 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1386 T + p^{3} T^{2} \) |
| 97 | \( 1 + 130 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.489339103197148158185701807466, −7.82200510857480130095675491104, −6.98702407686584684973630990008, −6.31255153739937207197465657630, −5.47919986680596515476883418946, −4.74979932933948890875611834490, −3.88891588361198760084411014626, −3.01377406472305668177120501580, −2.02924033501782736534904015036, −0.35911169289529144392631994048,
0.35911169289529144392631994048, 2.02924033501782736534904015036, 3.01377406472305668177120501580, 3.88891588361198760084411014626, 4.74979932933948890875611834490, 5.47919986680596515476883418946, 6.31255153739937207197465657630, 6.98702407686584684973630990008, 7.82200510857480130095675491104, 8.489339103197148158185701807466
