| L(s) = 1 | + 0.337·2-s + 2.44·3-s − 1.88·4-s + 0.824·6-s + 2.26·7-s − 1.31·8-s + 2.98·9-s − 4.20·11-s − 4.61·12-s + 1.32·13-s + 0.764·14-s + 3.33·16-s − 1.12·17-s + 1.00·18-s − 1.53·19-s + 5.54·21-s − 1.41·22-s − 5.10·23-s − 3.20·24-s + 0.447·26-s − 0.0377·27-s − 4.27·28-s − 5.78·29-s − 4.45·31-s + 3.74·32-s − 10.2·33-s − 0.377·34-s + ⋯ |
| L(s) = 1 | + 0.238·2-s + 1.41·3-s − 0.943·4-s + 0.336·6-s + 0.857·7-s − 0.463·8-s + 0.994·9-s − 1.26·11-s − 1.33·12-s + 0.367·13-s + 0.204·14-s + 0.832·16-s − 0.271·17-s + 0.237·18-s − 0.351·19-s + 1.21·21-s − 0.302·22-s − 1.06·23-s − 0.654·24-s + 0.0877·26-s − 0.00726·27-s − 0.808·28-s − 1.07·29-s − 0.800·31-s + 0.661·32-s − 1.78·33-s − 0.0648·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 0.337T + 2T^{2} \) |
| 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 - 2.26T + 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 + 5.78T + 29T^{2} \) |
| 31 | \( 1 + 4.45T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 + 8.63T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 0.172T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 7.54T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 1.82T + 73T^{2} \) |
| 79 | \( 1 + 4.65T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 3.91T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.919695410976790944324540486477, −7.49787409966193160377319575615, −6.14099975194830621788087350287, −5.43657361525719572635398976750, −4.64003875601243754654173705232, −4.02153844810971245641430413858, −3.27293092912384577756067598848, −2.43006970338031525233346845660, −1.61567535664578765677465022961, 0,
1.61567535664578765677465022961, 2.43006970338031525233346845660, 3.27293092912384577756067598848, 4.02153844810971245641430413858, 4.64003875601243754654173705232, 5.43657361525719572635398976750, 6.14099975194830621788087350287, 7.49787409966193160377319575615, 7.919695410976790944324540486477
