| L(s) = 1 | − 2.28·2-s − 1.65·3-s + 3.23·4-s − 1.16·5-s + 3.78·6-s − 2.98·7-s − 2.82·8-s − 0.258·9-s + 2.66·10-s + 1.34·11-s − 5.35·12-s + 6.84·14-s + 1.92·15-s + 0.00191·16-s + 6.67·17-s + 0.591·18-s − 2.81·19-s − 3.77·20-s + 4.95·21-s − 3.07·22-s − 3.43·23-s + 4.68·24-s − 3.64·25-s + 5.39·27-s − 9.67·28-s − 0.221·29-s − 4.41·30-s + ⋯ |
| L(s) = 1 | − 1.61·2-s − 0.955·3-s + 1.61·4-s − 0.521·5-s + 1.54·6-s − 1.13·7-s − 1.00·8-s − 0.0861·9-s + 0.843·10-s + 0.405·11-s − 1.54·12-s + 1.82·14-s + 0.498·15-s + 0.000477·16-s + 1.61·17-s + 0.139·18-s − 0.646·19-s − 0.843·20-s + 1.08·21-s − 0.656·22-s − 0.717·23-s + 0.956·24-s − 0.728·25-s + 1.03·27-s − 1.82·28-s − 0.0411·29-s − 0.806·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 + 1.65T + 3T^{2} \) |
| 5 | \( 1 + 1.16T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 17 | \( 1 - 6.67T + 17T^{2} \) |
| 19 | \( 1 + 2.81T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 + 0.221T + 29T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 4.81T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 + 6.66T + 47T^{2} \) |
| 53 | \( 1 - 4.26T + 53T^{2} \) |
| 59 | \( 1 + 2.22T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 - 4.81T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 5.01T + 79T^{2} \) |
| 83 | \( 1 + 3.24T + 83T^{2} \) |
| 89 | \( 1 + 9.01T + 89T^{2} \) |
| 97 | \( 1 + 4.77T + 97T^{2} \) |
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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
−8.086778116438562387687596409631, −7.09733562110478422647748788729, −6.65887456567754528934355761598, −5.94572840072789868650490381856, −5.21063907439210186707513397082, −3.93346178364468987666793209916, −3.19304507384507323999054873835, −1.97836201420848664392545534949, −0.792581011937069463642657322772, 0,
0.792581011937069463642657322772, 1.97836201420848664392545534949, 3.19304507384507323999054873835, 3.93346178364468987666793209916, 5.21063907439210186707513397082, 5.94572840072789868650490381856, 6.65887456567754528934355761598, 7.09733562110478422647748788729, 8.086778116438562387687596409631
