L(s) = 1 | + 2-s + 3.10·3-s + 4-s + 1.73·5-s + 3.10·6-s + 3.71·7-s + 8-s + 6.61·9-s + 1.73·10-s + 0.351·11-s + 3.10·12-s − 3.27·13-s + 3.71·14-s + 5.37·15-s + 16-s + 0.752·17-s + 6.61·18-s − 19-s + 1.73·20-s + 11.5·21-s + 0.351·22-s − 6.92·23-s + 3.10·24-s − 1.98·25-s − 3.27·26-s + 11.1·27-s + 3.71·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.78·3-s + 0.5·4-s + 0.776·5-s + 1.26·6-s + 1.40·7-s + 0.353·8-s + 2.20·9-s + 0.548·10-s + 0.105·11-s + 0.894·12-s − 0.909·13-s + 0.993·14-s + 1.38·15-s + 0.250·16-s + 0.182·17-s + 1.55·18-s − 0.229·19-s + 0.388·20-s + 2.51·21-s + 0.0748·22-s − 1.44·23-s + 0.632·24-s − 0.397·25-s − 0.643·26-s + 2.15·27-s + 0.702·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.930346515\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.930346515\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 11 | \( 1 - 0.351T + 11T^{2} \) |
| 13 | \( 1 + 3.27T + 13T^{2} \) |
| 17 | \( 1 - 0.752T + 17T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 1.99T + 29T^{2} \) |
| 31 | \( 1 + 1.76T + 31T^{2} \) |
| 37 | \( 1 + 9.46T + 37T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 - 4.84T + 43T^{2} \) |
| 47 | \( 1 - 9.66T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 2.58T + 59T^{2} \) |
| 61 | \( 1 + 3.16T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 4.14T + 73T^{2} \) |
| 79 | \( 1 - 2.86T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 1.69T + 97T^{2} \) |
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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
−7.84996096998405135753605297474, −7.40663936610516581720528412856, −6.49721171332429395224742117880, −5.59377296034726052123100839563, −4.85827599097144185838667517281, −4.19307864444928110051427074550, −3.54959528652984940189038288690, −2.44395838147375659093461683932, −2.14450465079399996780562938026, −1.43562708965236668339140880515,
1.43562708965236668339140880515, 2.14450465079399996780562938026, 2.44395838147375659093461683932, 3.54959528652984940189038288690, 4.19307864444928110051427074550, 4.85827599097144185838667517281, 5.59377296034726052123100839563, 6.49721171332429395224742117880, 7.40663936610516581720528412856, 7.84996096998405135753605297474