| L(s) = 1 | − 2·3-s − 5-s − 4·7-s + 9-s − 11-s + 4·13-s + 2·15-s + 4·19-s + 8·21-s + 6·23-s + 25-s + 4·27-s + 6·29-s − 8·31-s + 2·33-s + 4·35-s − 8·39-s + 6·41-s − 8·43-s − 45-s + 6·47-s + 9·49-s − 6·53-s + 55-s − 8·57-s + 12·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.516·15-s + 0.917·19-s + 1.74·21-s + 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 1.43·31-s + 0.348·33-s + 0.676·35-s − 1.28·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s + 0.875·47-s + 9/7·49-s − 0.824·53-s + 0.134·55-s − 1.05·57-s + 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.07893076051328, −12.59882647077194, −12.22587079831857, −11.77688083643415, −11.21067685737856, −11.02421575158807, −10.41203082734855, −10.11597221786042, −9.522196951694045, −8.911655682118551, −8.747471479011251, −8.016033442297515, −7.327607793711959, −6.996586770720735, −6.554478989657221, −6.096303481397097, −5.528769632739463, −5.378383636012697, −4.570158585389574, −4.058495662662019, −3.477667194863838, −2.907605075339168, −2.707813369702774, −1.282875539885425, −1.138904137249732, 0, 0,
1.138904137249732, 1.282875539885425, 2.707813369702774, 2.907605075339168, 3.477667194863838, 4.058495662662019, 4.570158585389574, 5.378383636012697, 5.528769632739463, 6.096303481397097, 6.554478989657221, 6.996586770720735, 7.327607793711959, 8.016033442297515, 8.747471479011251, 8.911655682118551, 9.522196951694045, 10.11597221786042, 10.41203082734855, 11.02421575158807, 11.21067685737856, 11.77688083643415, 12.22587079831857, 12.59882647077194, 13.07893076051328
