| L(s) = 1 | − 2·2-s + 9.15·3-s + 4·4-s + 5·5-s − 18.3·6-s − 8·8-s + 56.7·9-s − 10·10-s − 35.7·11-s + 36.6·12-s + 67.4·13-s + 45.7·15-s + 16·16-s − 19.1·17-s − 113.·18-s − 86.6·19-s + 20·20-s + 71.5·22-s + 195.·23-s − 73.2·24-s + 25·25-s − 134.·26-s + 272.·27-s + 272.·29-s − 91.5·30-s + 21.6·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.76·3-s + 0.5·4-s + 0.447·5-s − 1.24·6-s − 0.353·8-s + 2.10·9-s − 0.316·10-s − 0.980·11-s + 0.880·12-s + 1.43·13-s + 0.787·15-s + 0.250·16-s − 0.273·17-s − 1.48·18-s − 1.04·19-s + 0.223·20-s + 0.693·22-s + 1.76·23-s − 0.622·24-s + 0.200·25-s − 1.01·26-s + 1.94·27-s + 1.74·29-s − 0.556·30-s + 0.125·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.053028169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.053028169\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 9.15T + 27T^{2} \) |
| 11 | \( 1 + 35.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 67.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 195.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 272.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 21.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 67.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 107.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 609.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 645.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 140.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 834.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 491.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 479.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 610.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 153.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 784.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 227.T + 9.12e5T^{2} \) |
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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
−10.41087069029220862171590628833, −9.375910133658802552316749704325, −8.664707618009847400862718333884, −8.239811246963742367394729492564, −7.20395601366924217683032560285, −6.21638757286163002046904540263, −4.58113646335205581727739674422, −3.20760750191956727077538192907, −2.45545267727539309947318897130, −1.23492185518351840387013308070,
1.23492185518351840387013308070, 2.45545267727539309947318897130, 3.20760750191956727077538192907, 4.58113646335205581727739674422, 6.21638757286163002046904540263, 7.20395601366924217683032560285, 8.239811246963742367394729492564, 8.664707618009847400862718333884, 9.375910133658802552316749704325, 10.41087069029220862171590628833
