| L(s) = 1 | + 1.55·2-s − 0.910·3-s + 0.420·4-s − 4.33·5-s − 1.41·6-s − 1.62·7-s − 2.45·8-s − 2.17·9-s − 6.73·10-s + 11-s − 0.383·12-s + 2.03·13-s − 2.53·14-s + 3.94·15-s − 4.66·16-s − 0.468·17-s − 3.37·18-s + 3.87·19-s − 1.82·20-s + 1.48·21-s + 1.55·22-s − 0.764·23-s + 2.23·24-s + 13.7·25-s + 3.16·26-s + 4.70·27-s − 0.684·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 0.525·3-s + 0.210·4-s − 1.93·5-s − 0.578·6-s − 0.615·7-s − 0.868·8-s − 0.723·9-s − 2.13·10-s + 0.301·11-s − 0.110·12-s + 0.564·13-s − 0.677·14-s + 1.01·15-s − 1.16·16-s − 0.113·17-s − 0.796·18-s + 0.888·19-s − 0.407·20-s + 0.323·21-s + 0.331·22-s − 0.159·23-s + 0.456·24-s + 2.75·25-s + 0.620·26-s + 0.906·27-s − 0.129·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 3 | \( 1 + 0.910T + 3T^{2} \) |
| 5 | \( 1 + 4.33T + 5T^{2} \) |
| 7 | \( 1 + 1.62T + 7T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 + 0.468T + 17T^{2} \) |
| 19 | \( 1 - 3.87T + 19T^{2} \) |
| 23 | \( 1 + 0.764T + 23T^{2} \) |
| 31 | \( 1 - 4.94T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 + 9.27T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 - 8.28T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 8.40T + 59T^{2} \) |
| 61 | \( 1 - 2.97T + 61T^{2} \) |
| 67 | \( 1 + 5.68T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.11221102425309028063579524115, −6.62708784823067516366956438838, −5.86834544594476659098840359050, −5.21707820473077496394866557120, −4.45856977979095604832625255955, −3.93428909392475790900937934003, −3.25678175326646662160184693030, −2.81617544885669414562835550851, −0.870996564678981134147961925253, 0,
0.870996564678981134147961925253, 2.81617544885669414562835550851, 3.25678175326646662160184693030, 3.93428909392475790900937934003, 4.45856977979095604832625255955, 5.21707820473077496394866557120, 5.86834544594476659098840359050, 6.62708784823067516366956438838, 7.11221102425309028063579524115
