| L(s) = 1 | + 2·2-s + 6.40·3-s + 4·4-s + 5·5-s + 12.8·6-s + 8·8-s + 13.9·9-s + 10·10-s + 39.0·11-s + 25.6·12-s − 21.7·13-s + 32.0·15-s + 16·16-s + 12.1·17-s + 27.9·18-s + 86.5·19-s + 20·20-s + 78.0·22-s − 7.04·23-s + 51.2·24-s + 25·25-s − 43.4·26-s − 83.3·27-s + 57.8·29-s + 64.0·30-s + 241.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.23·3-s + 0.5·4-s + 0.447·5-s + 0.871·6-s + 0.353·8-s + 0.517·9-s + 0.316·10-s + 1.07·11-s + 0.615·12-s − 0.463·13-s + 0.550·15-s + 0.250·16-s + 0.173·17-s + 0.365·18-s + 1.04·19-s + 0.223·20-s + 0.756·22-s − 0.0638·23-s + 0.435·24-s + 0.200·25-s − 0.327·26-s − 0.594·27-s + 0.370·29-s + 0.389·30-s + 1.39·31-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
| \(L(2)\) |
\(\approx\) |
\(5.298220069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.298220069\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 6.40T + 27T^{2} \) |
| 11 | \( 1 - 39.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.04T + 1.21e4T^{2} \) |
| 29 | \( 1 - 57.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 396.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 26.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 44.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 501.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 717.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 455.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 533.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 814.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 258.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 839.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 130.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−10.42957861566807238555678385861, −9.532688524741518988885888945124, −8.834713753216469294486696818920, −7.79837625588544678015010652868, −6.89248281853094826341261653659, −5.84068575755927202902411024875, −4.65050197647841726785590422217, −3.52105614567346414943070004722, −2.68412498219589486544840127892, −1.46279573137808593249687044195,
1.46279573137808593249687044195, 2.68412498219589486544840127892, 3.52105614567346414943070004722, 4.65050197647841726785590422217, 5.84068575755927202902411024875, 6.89248281853094826341261653659, 7.79837625588544678015010652868, 8.834713753216469294486696818920, 9.532688524741518988885888945124, 10.42957861566807238555678385861
