| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 15·5-s − 6·6-s + 17·7-s − 8·8-s + 9·9-s − 30·10-s + 24·11-s + 12·12-s + 2·13-s − 34·14-s + 45·15-s + 16·16-s − 48·17-s − 18·18-s − 115·19-s + 60·20-s + 51·21-s − 48·22-s + 30·23-s − 24·24-s + 100·25-s − 4·26-s + 27·27-s + 68·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.917·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.657·11-s + 0.288·12-s + 0.0426·13-s − 0.649·14-s + 0.774·15-s + 1/4·16-s − 0.684·17-s − 0.235·18-s − 1.38·19-s + 0.670·20-s + 0.529·21-s − 0.465·22-s + 0.271·23-s − 0.204·24-s + 4/5·25-s − 0.0301·26-s + 0.192·27-s + 0.458·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.108418327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.108418327\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 31 | \( 1 - p T \) |
| good | 5 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 17 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 48 T + p^{3} T^{2} \) |
| 19 | \( 1 + 115 T + p^{3} T^{2} \) |
| 23 | \( 1 - 30 T + p^{3} T^{2} \) |
| 29 | \( 1 - 264 T + p^{3} T^{2} \) |
| 37 | \( 1 + 160 T + p^{3} T^{2} \) |
| 41 | \( 1 + 51 T + p^{3} T^{2} \) |
| 43 | \( 1 - 128 T + p^{3} T^{2} \) |
| 47 | \( 1 - 480 T + p^{3} T^{2} \) |
| 53 | \( 1 - 132 T + p^{3} T^{2} \) |
| 59 | \( 1 - 309 T + p^{3} T^{2} \) |
| 61 | \( 1 + 280 T + p^{3} T^{2} \) |
| 67 | \( 1 + 604 T + p^{3} T^{2} \) |
| 71 | \( 1 + 159 T + p^{3} T^{2} \) |
| 73 | \( 1 + 652 T + p^{3} T^{2} \) |
| 79 | \( 1 + 838 T + p^{3} T^{2} \) |
| 83 | \( 1 + 690 T + p^{3} T^{2} \) |
| 89 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 97 | \( 1 - 329 T + p^{3} T^{2} \) |
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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
−12.06495447893313175955387626698, −10.82069091600531217290705708112, −10.09028656170583999344167716292, −8.962535729199343705841388199387, −8.464451885243023668591093502051, −7.01428179093715877723063709011, −6.01704739615297812954969766833, −4.49153125823335119317257629826, −2.48304245409309856227462379789, −1.46869518310948374825592367895,
1.46869518310948374825592367895, 2.48304245409309856227462379789, 4.49153125823335119317257629826, 6.01704739615297812954969766833, 7.01428179093715877723063709011, 8.464451885243023668591093502051, 8.962535729199343705841388199387, 10.09028656170583999344167716292, 10.82069091600531217290705708112, 12.06495447893313175955387626698
