| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 15·5-s − 6·6-s − 8·8-s + 9·9-s − 30·10-s − 9·11-s + 12·12-s + 88·13-s + 45·15-s + 16·16-s + 84·17-s − 18·18-s − 104·19-s + 60·20-s + 18·22-s − 84·23-s − 24·24-s + 100·25-s − 176·26-s + 27·27-s + 51·29-s − 90·30-s − 185·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.246·11-s + 0.288·12-s + 1.87·13-s + 0.774·15-s + 1/4·16-s + 1.19·17-s − 0.235·18-s − 1.25·19-s + 0.670·20-s + 0.174·22-s − 0.761·23-s − 0.204·24-s + 4/5·25-s − 1.32·26-s + 0.192·27-s + 0.326·29-s − 0.547·30-s − 1.07·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.215343586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.215343586\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 9 T + p^{3} T^{2} \) |
| 13 | \( 1 - 88 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 + 104 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 51 T + p^{3} T^{2} \) |
| 31 | \( 1 + 185 T + p^{3} T^{2} \) |
| 37 | \( 1 - 44 T + p^{3} T^{2} \) |
| 41 | \( 1 - 168 T + p^{3} T^{2} \) |
| 43 | \( 1 - 326 T + p^{3} T^{2} \) |
| 47 | \( 1 - 138 T + p^{3} T^{2} \) |
| 53 | \( 1 - 639 T + p^{3} T^{2} \) |
| 59 | \( 1 + 159 T + p^{3} T^{2} \) |
| 61 | \( 1 + 722 T + p^{3} T^{2} \) |
| 67 | \( 1 + 166 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1086 T + p^{3} T^{2} \) |
| 73 | \( 1 + 218 T + p^{3} T^{2} \) |
| 79 | \( 1 + 583 T + p^{3} T^{2} \) |
| 83 | \( 1 - 597 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 - 169 T + p^{3} T^{2} \) |
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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
−10.88529348848956971258729385265, −10.33291432654875682253154571192, −9.349990599215708836169167255400, −8.662544510915004589071555714184, −7.71774820456883591125177336889, −6.33214763357121731801833094375, −5.69946738134048343284850012256, −3.80208363316279465844656945056, −2.36375563430464180898764447650, −1.27299364149994760826847974923,
1.27299364149994760826847974923, 2.36375563430464180898764447650, 3.80208363316279465844656945056, 5.69946738134048343284850012256, 6.33214763357121731801833094375, 7.71774820456883591125177336889, 8.662544510915004589071555714184, 9.349990599215708836169167255400, 10.33291432654875682253154571192, 10.88529348848956971258729385265
