L(s) = 1 | + 2-s − 1.69·3-s + 4-s − 3.34·5-s − 1.69·6-s + 0.398·7-s + 8-s − 0.140·9-s − 3.34·10-s + 2.98·11-s − 1.69·12-s + 4.90·13-s + 0.398·14-s + 5.65·15-s + 16-s − 7.88·17-s − 0.140·18-s + 7.38·19-s − 3.34·20-s − 0.674·21-s + 2.98·22-s − 23-s − 1.69·24-s + 6.19·25-s + 4.90·26-s + 5.31·27-s + 0.398·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.976·3-s + 0.5·4-s − 1.49·5-s − 0.690·6-s + 0.150·7-s + 0.353·8-s − 0.0467·9-s − 1.05·10-s + 0.898·11-s − 0.488·12-s + 1.35·13-s + 0.106·14-s + 1.46·15-s + 0.250·16-s − 1.91·17-s − 0.0330·18-s + 1.69·19-s − 0.748·20-s − 0.147·21-s + 0.635·22-s − 0.208·23-s − 0.345·24-s + 1.23·25-s + 0.961·26-s + 1.02·27-s + 0.0753·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.402317211\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.402317211\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 1.69T + 3T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 - 0.398T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 - 4.90T + 13T^{2} \) |
| 17 | \( 1 + 7.88T + 17T^{2} \) |
| 19 | \( 1 - 7.38T + 19T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 + 6.81T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 2.44T + 47T^{2} \) |
| 53 | \( 1 - 9.95T + 53T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 + 9.23T + 67T^{2} \) |
| 71 | \( 1 - 8.52T + 71T^{2} \) |
| 73 | \( 1 - 2.01T + 73T^{2} \) |
| 79 | \( 1 - 6.10T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 2.06T + 97T^{2} \) |
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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
−7.949247711837089913696960267848, −7.05849931680260030434638671594, −6.70008868852340112527550607402, −5.86550331968046599117646025349, −5.17697702448869494590459012559, −4.41150058779545837954659802219, −3.75049518051160805853179879008, −3.23498073416861488181426116018, −1.74331588587805916131298773136, −0.59447199728121113332842566446,
0.59447199728121113332842566446, 1.74331588587805916131298773136, 3.23498073416861488181426116018, 3.75049518051160805853179879008, 4.41150058779545837954659802219, 5.17697702448869494590459012559, 5.86550331968046599117646025349, 6.70008868852340112527550607402, 7.05849931680260030434638671594, 7.949247711837089913696960267848