| L(s) = 1 | + 2-s + 1.90·3-s + 4-s − 5-s + 1.90·6-s − 3.95·7-s + 8-s + 0.622·9-s − 10-s + 0.890·11-s + 1.90·12-s + 13-s − 3.95·14-s − 1.90·15-s + 16-s + 6.51·17-s + 0.622·18-s − 4.03·19-s − 20-s − 7.52·21-s + 0.890·22-s + 6.24·23-s + 1.90·24-s + 25-s + 26-s − 4.52·27-s − 3.95·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.09·3-s + 0.5·4-s − 0.447·5-s + 0.777·6-s − 1.49·7-s + 0.353·8-s + 0.207·9-s − 0.316·10-s + 0.268·11-s + 0.549·12-s + 0.277·13-s − 1.05·14-s − 0.491·15-s + 0.250·16-s + 1.58·17-s + 0.146·18-s − 0.924·19-s − 0.223·20-s − 1.64·21-s + 0.189·22-s + 1.30·23-s + 0.388·24-s + 0.200·25-s + 0.196·26-s − 0.870·27-s − 0.747·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.668207010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.668207010\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 1.90T + 3T^{2} \) |
| 7 | \( 1 + 3.95T + 7T^{2} \) |
| 11 | \( 1 - 0.890T + 11T^{2} \) |
| 17 | \( 1 - 6.51T + 17T^{2} \) |
| 19 | \( 1 + 4.03T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 - 4.15T + 29T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 4.78T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 4.43T + 61T^{2} \) |
| 67 | \( 1 - 4.30T + 67T^{2} \) |
| 71 | \( 1 - 6.03T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 - 1.27T + 79T^{2} \) |
| 83 | \( 1 - 2.95T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.361280904344167372518081996944, −7.70475757361847868516443350781, −6.97261254668465922079748034785, −6.22002020716822372292619515093, −5.56963881428917035771326804488, −4.34113451090155009440928042068, −3.68957745540379648586451291668, −3.02364921360726517540221728496, −2.54096078120525341099629277275, −0.944224929106252213037306613977,
0.944224929106252213037306613977, 2.54096078120525341099629277275, 3.02364921360726517540221728496, 3.68957745540379648586451291668, 4.34113451090155009440928042068, 5.56963881428917035771326804488, 6.22002020716822372292619515093, 6.97261254668465922079748034785, 7.70475757361847868516443350781, 8.361280904344167372518081996944
