| L(s) = 1 | − 2-s − 0.442·3-s + 4-s − 0.891·5-s + 0.442·6-s + 2.52·7-s − 8-s − 2.80·9-s + 0.891·10-s + 1.95·11-s − 0.442·12-s − 6.45·13-s − 2.52·14-s + 0.394·15-s + 16-s − 3.42·17-s + 2.80·18-s − 0.891·20-s − 1.11·21-s − 1.95·22-s + 8.18·23-s + 0.442·24-s − 4.20·25-s + 6.45·26-s + 2.56·27-s + 2.52·28-s + 4.58·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.255·3-s + 0.5·4-s − 0.398·5-s + 0.180·6-s + 0.952·7-s − 0.353·8-s − 0.934·9-s + 0.281·10-s + 0.588·11-s − 0.127·12-s − 1.79·13-s − 0.673·14-s + 0.101·15-s + 0.250·16-s − 0.830·17-s + 0.660·18-s − 0.199·20-s − 0.243·21-s − 0.416·22-s + 1.70·23-s + 0.0903·24-s − 0.841·25-s + 1.26·26-s + 0.494·27-s + 0.476·28-s + 0.850·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.442T + 3T^{2} \) |
| 5 | \( 1 + 0.891T + 5T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 + 6.45T + 13T^{2} \) |
| 17 | \( 1 + 3.42T + 17T^{2} \) |
| 23 | \( 1 - 8.18T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 + 8.79T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 + 3.48T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 3.76T + 53T^{2} \) |
| 59 | \( 1 - 2.84T + 59T^{2} \) |
| 61 | \( 1 + 2.45T + 61T^{2} \) |
| 67 | \( 1 + 2.67T + 67T^{2} \) |
| 71 | \( 1 - 0.0564T + 71T^{2} \) |
| 73 | \( 1 + 6.96T + 73T^{2} \) |
| 79 | \( 1 + 9.13T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 1.86T + 89T^{2} \) |
| 97 | \( 1 - 7.82T + 97T^{2} \) |
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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
−9.951180960234514291668249800819, −8.981307476924623645647470360089, −8.389014342844171384415917650736, −7.38862157008210743689981400887, −6.74875363488224427077564734796, −5.38055937288582420786171309552, −4.65497656888459602400858790997, −3.09734071153305472536600502162, −1.83950432187501078939883776647, 0,
1.83950432187501078939883776647, 3.09734071153305472536600502162, 4.65497656888459602400858790997, 5.38055937288582420786171309552, 6.74875363488224427077564734796, 7.38862157008210743689981400887, 8.389014342844171384415917650736, 8.981307476924623645647470360089, 9.951180960234514291668249800819
