| L(s) = 1 | − 2.39·2-s − 2.59·3-s + 3.74·4-s + 5-s + 6.22·6-s − 2.89·7-s − 4.19·8-s + 3.73·9-s − 2.39·10-s − 11-s − 9.72·12-s + 4.73·13-s + 6.93·14-s − 2.59·15-s + 2.56·16-s − 5.65·17-s − 8.94·18-s + 19-s + 3.74·20-s + 7.50·21-s + 2.39·22-s − 4.00·23-s + 10.8·24-s + 25-s − 11.3·26-s − 1.89·27-s − 10.8·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s − 1.49·3-s + 1.87·4-s + 0.447·5-s + 2.53·6-s − 1.09·7-s − 1.48·8-s + 1.24·9-s − 0.758·10-s − 0.301·11-s − 2.80·12-s + 1.31·13-s + 1.85·14-s − 0.669·15-s + 0.640·16-s − 1.37·17-s − 2.10·18-s + 0.229·19-s + 0.838·20-s + 1.63·21-s + 0.511·22-s − 0.835·23-s + 2.22·24-s + 0.200·25-s − 2.22·26-s − 0.364·27-s − 2.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 3 | \( 1 + 2.59T + 3T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 23 | \( 1 + 4.00T + 23T^{2} \) |
| 29 | \( 1 - 9.32T + 29T^{2} \) |
| 31 | \( 1 + 6.60T + 31T^{2} \) |
| 37 | \( 1 - 6.07T + 37T^{2} \) |
| 41 | \( 1 - 5.47T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 0.295T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 + 5.54T + 59T^{2} \) |
| 61 | \( 1 + 1.01T + 61T^{2} \) |
| 67 | \( 1 - 6.98T + 67T^{2} \) |
| 71 | \( 1 + 1.02T + 71T^{2} \) |
| 73 | \( 1 + 0.202T + 73T^{2} \) |
| 79 | \( 1 - 7.28T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 4.81T + 97T^{2} \) |
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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
−9.549681294916700164901128603519, −8.934289856729418910706133468694, −7.940648506323679562575967370187, −6.78980677918249161271724970366, −6.36370934179019048522105382055, −5.72780243667146431756540116883, −4.27756842017052577651789645402, −2.60532323274751192430795141756, −1.15756315766356415567357572527, 0,
1.15756315766356415567357572527, 2.60532323274751192430795141756, 4.27756842017052577651789645402, 5.72780243667146431756540116883, 6.36370934179019048522105382055, 6.78980677918249161271724970366, 7.940648506323679562575967370187, 8.934289856729418910706133468694, 9.549681294916700164901128603519
