L(s) = 1 | + 0.999·3-s − 5-s − 7-s − 2.00·9-s + 3.49·11-s − 4.35·13-s − 0.999·15-s + 8.07·17-s − 0.849·19-s − 0.999·21-s − 3.31·23-s + 25-s − 4.99·27-s − 7.81·29-s − 7.69·31-s + 3.49·33-s + 35-s + 8.99·37-s − 4.34·39-s + 4.53·41-s − 43-s + 2.00·45-s + 11.1·47-s + 49-s + 8.06·51-s + 13.0·53-s − 3.49·55-s + ⋯ |
L(s) = 1 | + 0.576·3-s − 0.447·5-s − 0.377·7-s − 0.667·9-s + 1.05·11-s − 1.20·13-s − 0.257·15-s + 1.95·17-s − 0.194·19-s − 0.218·21-s − 0.691·23-s + 0.200·25-s − 0.961·27-s − 1.45·29-s − 1.38·31-s + 0.608·33-s + 0.169·35-s + 1.47·37-s − 0.696·39-s + 0.708·41-s − 0.152·43-s + 0.298·45-s + 1.63·47-s + 0.142·49-s + 1.12·51-s + 1.79·53-s − 0.471·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.811168634\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.811168634\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 - 0.999T + 3T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 17 | \( 1 - 8.07T + 17T^{2} \) |
| 19 | \( 1 + 0.849T + 19T^{2} \) |
| 23 | \( 1 + 3.31T + 23T^{2} \) |
| 29 | \( 1 + 7.81T + 29T^{2} \) |
| 31 | \( 1 + 7.69T + 31T^{2} \) |
| 37 | \( 1 - 8.99T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 8.29T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 3.32T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 2.52T + 79T^{2} \) |
| 83 | \( 1 - 1.42T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 1.38T + 97T^{2} \) |
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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
−7.84044569106966645070942289251, −7.62589269633043400392491785722, −6.82148354174937621196147231466, −5.74278507793596119406164041811, −5.45561530057163474715880840886, −4.09085270536948886153008104047, −3.71195769140304314400795883723, −2.84112710296438826195622475876, −2.00767312522748582134982069213, −0.67135157522577892639996708316,
0.67135157522577892639996708316, 2.00767312522748582134982069213, 2.84112710296438826195622475876, 3.71195769140304314400795883723, 4.09085270536948886153008104047, 5.45561530057163474715880840886, 5.74278507793596119406164041811, 6.82148354174937621196147231466, 7.62589269633043400392491785722, 7.84044569106966645070942289251