| L(s) = 1 | − 1.91·2-s + 0.294·3-s + 1.68·4-s − 5-s − 0.566·6-s − 7-s + 0.604·8-s − 2.91·9-s + 1.91·10-s + 2.97·11-s + 0.497·12-s + 3.42·13-s + 1.91·14-s − 0.294·15-s − 4.53·16-s + 0.598·17-s + 5.59·18-s − 5.23·19-s − 1.68·20-s − 0.294·21-s − 5.71·22-s − 3.86·23-s + 0.178·24-s + 25-s − 6.58·26-s − 1.74·27-s − 1.68·28-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.170·3-s + 0.842·4-s − 0.447·5-s − 0.231·6-s − 0.377·7-s + 0.213·8-s − 0.970·9-s + 0.607·10-s + 0.896·11-s + 0.143·12-s + 0.951·13-s + 0.513·14-s − 0.0761·15-s − 1.13·16-s + 0.145·17-s + 1.31·18-s − 1.20·19-s − 0.376·20-s − 0.0643·21-s − 1.21·22-s − 0.806·23-s + 0.0364·24-s + 0.200·25-s − 1.29·26-s − 0.335·27-s − 0.318·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 1.91T + 2T^{2} \) |
| 3 | \( 1 - 0.294T + 3T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 13 | \( 1 - 3.42T + 13T^{2} \) |
| 17 | \( 1 - 0.598T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 + 3.86T + 23T^{2} \) |
| 29 | \( 1 - 7.39T + 29T^{2} \) |
| 31 | \( 1 + 3.91T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 5.62T + 41T^{2} \) |
| 43 | \( 1 + 2.16T + 43T^{2} \) |
| 47 | \( 1 + 2.98T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 - 3.76T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 4.69T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 8.01T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.941108669345453170141160504658, −6.74032712323604149402466082846, −6.50050684360321463256389969761, −5.63920426822470522453754646763, −4.45560724595448581176228197348, −3.86593341364007643214262069202, −2.94391555765529798890058483030, −1.98881416116781440224972405131, −0.998036484042610431023621046168, 0,
0.998036484042610431023621046168, 1.98881416116781440224972405131, 2.94391555765529798890058483030, 3.86593341364007643214262069202, 4.45560724595448581176228197348, 5.63920426822470522453754646763, 6.50050684360321463256389969761, 6.74032712323604149402466082846, 7.941108669345453170141160504658
