L(s) = 1 | + 2-s + 4-s + 5-s − 4.23·7-s + 8-s + 10-s + 2.01·11-s − 3.86·13-s − 4.23·14-s + 16-s + 6.23·17-s − 7.49·19-s + 20-s + 2.01·22-s + 5.86·23-s + 25-s − 3.86·26-s − 4.23·28-s + 0.375·29-s + 9.47·31-s + 32-s + 6.23·34-s − 4.23·35-s + 37-s − 7.49·38-s + 40-s + 6.72·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.60·7-s + 0.353·8-s + 0.316·10-s + 0.608·11-s − 1.07·13-s − 1.13·14-s + 0.250·16-s + 1.51·17-s − 1.71·19-s + 0.223·20-s + 0.430·22-s + 1.22·23-s + 0.200·25-s − 0.757·26-s − 0.800·28-s + 0.0696·29-s + 1.70·31-s + 0.176·32-s + 1.06·34-s − 0.716·35-s + 0.164·37-s − 1.21·38-s + 0.158·40-s + 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.713309698\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.713309698\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 2.01T + 11T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 - 6.23T + 17T^{2} \) |
| 19 | \( 1 + 7.49T + 19T^{2} \) |
| 23 | \( 1 - 5.86T + 23T^{2} \) |
| 29 | \( 1 - 0.375T + 29T^{2} \) |
| 31 | \( 1 - 9.47T + 31T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 - 8.72T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 0.861T + 53T^{2} \) |
| 59 | \( 1 + 4.03T + 59T^{2} \) |
| 61 | \( 1 - 4.12T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 6.59T + 97T^{2} \) |
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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
−8.808280261238214053156808580255, −7.60335154359358407216552059806, −6.96235639524964945620278182897, −6.21398041927424885979402426850, −5.81295540891992353102397531477, −4.71959102058278242584747052815, −3.96023091781569812895884972124, −2.97526942731867119785507463673, −2.44814425585896048930413788408, −0.881136298997573516803859375059,
0.881136298997573516803859375059, 2.44814425585896048930413788408, 2.97526942731867119785507463673, 3.96023091781569812895884972124, 4.71959102058278242584747052815, 5.81295540891992353102397531477, 6.21398041927424885979402426850, 6.96235639524964945620278182897, 7.60335154359358407216552059806, 8.808280261238214053156808580255