| L(s) = 1 | + 4·2-s + 26·3-s + 16·4-s + 104·6-s + 22·7-s + 64·8-s + 433·9-s − 768·11-s + 416·12-s + 46·13-s + 88·14-s + 256·16-s − 378·17-s + 1.73e3·18-s + 1.10e3·19-s + 572·21-s − 3.07e3·22-s + 1.98e3·23-s + 1.66e3·24-s + 184·26-s + 4.94e3·27-s + 352·28-s − 5.61e3·29-s − 3.98e3·31-s + 1.02e3·32-s − 1.99e4·33-s − 1.51e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.66·3-s + 1/2·4-s + 1.17·6-s + 0.169·7-s + 0.353·8-s + 1.78·9-s − 1.91·11-s + 0.833·12-s + 0.0754·13-s + 0.119·14-s + 1/4·16-s − 0.317·17-s + 1.25·18-s + 0.699·19-s + 0.283·21-s − 1.35·22-s + 0.782·23-s + 0.589·24-s + 0.0533·26-s + 1.30·27-s + 0.0848·28-s − 1.23·29-s − 0.745·31-s + 0.176·32-s − 3.19·33-s − 0.224·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.859803435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.859803435\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 26 T + p^{5} T^{2} \) |
| 7 | \( 1 - 22 T + p^{5} T^{2} \) |
| 11 | \( 1 + 768 T + p^{5} T^{2} \) |
| 13 | \( 1 - 46 T + p^{5} T^{2} \) |
| 17 | \( 1 + 378 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1100 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1986 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5610 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3988 T + p^{5} T^{2} \) |
| 37 | \( 1 - 142 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1542 T + p^{5} T^{2} \) |
| 43 | \( 1 - 5026 T + p^{5} T^{2} \) |
| 47 | \( 1 + 24738 T + p^{5} T^{2} \) |
| 53 | \( 1 - 14166 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28380 T + p^{5} T^{2} \) |
| 61 | \( 1 - 5522 T + p^{5} T^{2} \) |
| 67 | \( 1 - 24742 T + p^{5} T^{2} \) |
| 71 | \( 1 - 42372 T + p^{5} T^{2} \) |
| 73 | \( 1 - 52126 T + p^{5} T^{2} \) |
| 79 | \( 1 + 39640 T + p^{5} T^{2} \) |
| 83 | \( 1 - 59826 T + p^{5} T^{2} \) |
| 89 | \( 1 - 57690 T + p^{5} T^{2} \) |
| 97 | \( 1 - 144382 T + p^{5} T^{2} \) |
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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
−14.52225701533708896050067772796, −13.37909664764737732802977222766, −12.90378539784589334649273375895, −11.00687264443695052251940716717, −9.644537921633680114237411191234, −8.234555331278993564207697151367, −7.33022130699774262738792069522, −5.14211270114479246595325699305, −3.42340117537175849821087259903, −2.24804953198617332104216478548,
2.24804953198617332104216478548, 3.42340117537175849821087259903, 5.14211270114479246595325699305, 7.33022130699774262738792069522, 8.234555331278993564207697151367, 9.644537921633680114237411191234, 11.00687264443695052251940716717, 12.90378539784589334649273375895, 13.37909664764737732802977222766, 14.52225701533708896050067772796
