| L(s) = 1 | − 2·2-s − 6.85·3-s + 4·4-s + 13.7·6-s − 30.5·7-s − 8·8-s + 19.9·9-s + 20.3·11-s − 27.4·12-s − 3.87·13-s + 61.1·14-s + 16·16-s − 17·17-s − 39.8·18-s − 34.2·19-s + 209.·21-s − 40.7·22-s + 52.6·23-s + 54.8·24-s + 7.75·26-s + 48.3·27-s − 122.·28-s + 295.·29-s − 226.·31-s − 32·32-s − 139.·33-s + 34·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.31·3-s + 0.5·4-s + 0.932·6-s − 1.65·7-s − 0.353·8-s + 0.738·9-s + 0.558·11-s − 0.659·12-s − 0.0827·13-s + 1.16·14-s + 0.250·16-s − 0.242·17-s − 0.522·18-s − 0.413·19-s + 2.17·21-s − 0.394·22-s + 0.477·23-s + 0.466·24-s + 0.0585·26-s + 0.344·27-s − 0.825·28-s + 1.89·29-s − 1.31·31-s − 0.176·32-s − 0.735·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 + 6.85T + 27T^{2} \) |
| 7 | \( 1 + 30.5T + 343T^{2} \) |
| 11 | \( 1 - 20.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.87T + 2.19e3T^{2} \) |
| 19 | \( 1 + 34.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 52.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 295.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 226.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 32.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 50.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 41.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 550.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 355.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 49.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 795.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 30.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 820.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 572.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 232.T + 9.12e5T^{2} \) |
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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
−9.383684190091891202338838470764, −8.847978202829035761475721743608, −7.43805338910595115661984246604, −6.48955132685586221677160335741, −6.29516045542642271799623556044, −5.16629213209742242117548085194, −3.85445071525876810963475916227, −2.64871847179101110117191636409, −0.937612055558706986321975168576, 0,
0.937612055558706986321975168576, 2.64871847179101110117191636409, 3.85445071525876810963475916227, 5.16629213209742242117548085194, 6.29516045542642271799623556044, 6.48955132685586221677160335741, 7.43805338910595115661984246604, 8.847978202829035761475721743608, 9.383684190091891202338838470764
