| L(s) = 1 | − 2.41·2-s + 3.82·4-s − 2·5-s + 7-s − 4.41·8-s + 4.82·10-s + 0.828·11-s + 2·13-s − 2.41·14-s + 2.99·16-s + 2·17-s − 19-s − 7.65·20-s − 1.99·22-s − 0.828·23-s − 25-s − 4.82·26-s + 3.82·28-s + 1.17·29-s + 4·31-s + 1.58·32-s − 4.82·34-s − 2·35-s − 3.65·37-s + 2.41·38-s + 8.82·40-s − 3.17·41-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.91·4-s − 0.894·5-s + 0.377·7-s − 1.56·8-s + 1.52·10-s + 0.249·11-s + 0.554·13-s − 0.645·14-s + 0.749·16-s + 0.485·17-s − 0.229·19-s − 1.71·20-s − 0.426·22-s − 0.172·23-s − 0.200·25-s − 0.946·26-s + 0.723·28-s + 0.217·29-s + 0.718·31-s + 0.280·32-s − 0.828·34-s − 0.338·35-s − 0.601·37-s + 0.391·38-s + 1.39·40-s − 0.495·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5952693544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5952693544\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 3.65T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−9.754540148871096877924651508361, −8.656017805116165239425085543624, −8.369685943797747321476913177203, −7.53700858005512005645345162457, −6.88204577691985339651385259809, −5.85316340100579703043124872102, −4.46330878806533582085496261448, −3.35320145066424040991691564756, −1.96864415524163815882775272782, −0.75133810446997368040563145778,
0.75133810446997368040563145778, 1.96864415524163815882775272782, 3.35320145066424040991691564756, 4.46330878806533582085496261448, 5.85316340100579703043124872102, 6.88204577691985339651385259809, 7.53700858005512005645345162457, 8.369685943797747321476913177203, 8.656017805116165239425085543624, 9.754540148871096877924651508361
