| L(s) = 1 | + 2·2-s + 2·3-s + 4·4-s + 4·6-s + 16·7-s + 8·8-s − 23·9-s + 11·11-s + 8·12-s + 37·13-s + 32·14-s + 16·16-s + 36·17-s − 46·18-s + 5·19-s + 32·21-s + 22·22-s + 87·23-s + 16·24-s + 74·26-s − 100·27-s + 64·28-s + 45·29-s + 167·31-s + 32·32-s + 22·33-s + 72·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.384·3-s + 1/2·4-s + 0.272·6-s + 0.863·7-s + 0.353·8-s − 0.851·9-s + 0.301·11-s + 0.192·12-s + 0.789·13-s + 0.610·14-s + 1/4·16-s + 0.513·17-s − 0.602·18-s + 0.0603·19-s + 0.332·21-s + 0.213·22-s + 0.788·23-s + 0.136·24-s + 0.558·26-s − 0.712·27-s + 0.431·28-s + 0.288·29-s + 0.967·31-s + 0.176·32-s + 0.116·33-s + 0.363·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.052331364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.052331364\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 - 37 T + p^{3} T^{2} \) |
| 17 | \( 1 - 36 T + p^{3} T^{2} \) |
| 19 | \( 1 - 5 T + p^{3} T^{2} \) |
| 23 | \( 1 - 87 T + p^{3} T^{2} \) |
| 29 | \( 1 - 45 T + p^{3} T^{2} \) |
| 31 | \( 1 - 167 T + p^{3} T^{2} \) |
| 37 | \( 1 - 196 T + p^{3} T^{2} \) |
| 41 | \( 1 - 72 T + p^{3} T^{2} \) |
| 43 | \( 1 + 233 T + p^{3} T^{2} \) |
| 47 | \( 1 - 336 T + p^{3} T^{2} \) |
| 53 | \( 1 + 78 T + p^{3} T^{2} \) |
| 59 | \( 1 + 720 T + p^{3} T^{2} \) |
| 61 | \( 1 - 482 T + p^{3} T^{2} \) |
| 67 | \( 1 - 166 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1137 T + p^{3} T^{2} \) |
| 73 | \( 1 + 308 T + p^{3} T^{2} \) |
| 79 | \( 1 + 160 T + p^{3} T^{2} \) |
| 83 | \( 1 + 813 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1155 T + p^{3} T^{2} \) |
| 97 | \( 1 + 299 T + p^{3} T^{2} \) |
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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
−10.66549987268283273064356207248, −9.420478260920065726433394961507, −8.439489137236327858679655115808, −7.82212412482357628655027254615, −6.58759343164760590824341573690, −5.66310835727941098619585575156, −4.72423558051504695274317121298, −3.60037820681019830543435603488, −2.57984890714859215866876112412, −1.19630817976044843110812673907,
1.19630817976044843110812673907, 2.57984890714859215866876112412, 3.60037820681019830543435603488, 4.72423558051504695274317121298, 5.66310835727941098619585575156, 6.58759343164760590824341573690, 7.82212412482357628655027254615, 8.439489137236327858679655115808, 9.420478260920065726433394961507, 10.66549987268283273064356207248
