| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s − 2·9-s − 11-s + 12-s − 6·13-s − 14-s + 16-s − 5·17-s − 2·18-s − 7·19-s − 21-s − 22-s + 2·23-s + 24-s − 6·26-s − 5·27-s − 28-s + 8·29-s − 2·31-s + 32-s − 33-s − 5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.301·11-s + 0.288·12-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.471·18-s − 1.60·19-s − 0.218·21-s − 0.213·22-s + 0.417·23-s + 0.204·24-s − 1.17·26-s − 0.962·27-s − 0.188·28-s + 1.48·29-s − 0.359·31-s + 0.176·32-s − 0.174·33-s − 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 293 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.30881010986694, −13.79187980413389, −13.23535120686211, −12.74701441020343, −12.46990435756115, −11.89010764052409, −11.32677084283394, −10.85676259355520, −10.29796681110328, −9.860210337358988, −9.248919513744952, −8.605083191750698, −8.379613948065967, −7.715175927104766, −7.044239054075099, −6.583141668556754, −6.306540831579517, −5.371610856387037, −4.926296278431448, −4.552089162188338, −3.819270005094034, −3.137772659896159, −2.700651157352698, −2.202411586898299, −1.633715033216642, 0, 0,
1.633715033216642, 2.202411586898299, 2.700651157352698, 3.137772659896159, 3.819270005094034, 4.552089162188338, 4.926296278431448, 5.371610856387037, 6.306540831579517, 6.583141668556754, 7.044239054075099, 7.715175927104766, 8.379613948065967, 8.605083191750698, 9.248919513744952, 9.860210337358988, 10.29796681110328, 10.85676259355520, 11.32677084283394, 11.89010764052409, 12.46990435756115, 12.74701441020343, 13.23535120686211, 13.79187980413389, 14.30881010986694
