L(s) = 1 | + 2-s − 1.57·3-s + 4-s + 0.329·5-s − 1.57·6-s + 7-s + 8-s − 0.507·9-s + 0.329·10-s + 3.25·11-s − 1.57·12-s − 3.60·13-s + 14-s − 0.520·15-s + 16-s − 2.15·17-s − 0.507·18-s + 0.329·20-s − 1.57·21-s + 3.25·22-s + 3.48·23-s − 1.57·24-s − 4.89·25-s − 3.60·26-s + 5.53·27-s + 28-s − 7.27·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.911·3-s + 0.5·4-s + 0.147·5-s − 0.644·6-s + 0.377·7-s + 0.353·8-s − 0.169·9-s + 0.104·10-s + 0.980·11-s − 0.455·12-s − 0.999·13-s + 0.267·14-s − 0.134·15-s + 0.250·16-s − 0.521·17-s − 0.119·18-s + 0.0736·20-s − 0.344·21-s + 0.693·22-s + 0.727·23-s − 0.322·24-s − 0.978·25-s − 0.706·26-s + 1.06·27-s + 0.188·28-s − 1.35·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 1.57T + 3T^{2} \) |
| 5 | \( 1 - 0.329T + 5T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + 3.60T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 + 7.27T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + 8.60T + 37T^{2} \) |
| 41 | \( 1 - 0.542T + 41T^{2} \) |
| 43 | \( 1 + 5.42T + 43T^{2} \) |
| 47 | \( 1 - 2.57T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 9.47T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 + 0.375T + 67T^{2} \) |
| 71 | \( 1 + 6.95T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 + 5.46T + 83T^{2} \) |
| 89 | \( 1 + 0.179T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.64899476147587962847758654526, −6.86663757803141633934305979662, −6.40838305366844423617436759711, −5.49829633904466126505775409488, −5.11204609705626568770222789517, −4.29778832188017775177814702068, −3.47254348595272736487086856841, −2.38690642947636942060698843032, −1.46192001994720437650703499600, 0,
1.46192001994720437650703499600, 2.38690642947636942060698843032, 3.47254348595272736487086856841, 4.29778832188017775177814702068, 5.11204609705626568770222789517, 5.49829633904466126505775409488, 6.40838305366844423617436759711, 6.86663757803141633934305979662, 7.64899476147587962847758654526