| L(s) = 1 | − 2·2-s + 5.24·3-s + 4·4-s − 5·5-s − 10.4·6-s − 8·8-s + 0.485·9-s + 10·10-s − 1.68·11-s + 20.9·12-s − 38.2·13-s − 26.2·15-s + 16·16-s + 68.9·17-s − 0.970·18-s + 40.6·19-s − 20·20-s + 3.37·22-s + 120.·23-s − 41.9·24-s + 25·25-s + 76.5·26-s − 139.·27-s + 78.8·29-s + 52.4·30-s + 229.·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.00·3-s + 0.5·4-s − 0.447·5-s − 0.713·6-s − 0.353·8-s + 0.0179·9-s + 0.316·10-s − 0.0462·11-s + 0.504·12-s − 0.816·13-s − 0.451·15-s + 0.250·16-s + 0.983·17-s − 0.0127·18-s + 0.490·19-s − 0.223·20-s + 0.0326·22-s + 1.09·23-s − 0.356·24-s + 0.200·25-s + 0.577·26-s − 0.990·27-s + 0.504·29-s + 0.319·30-s + 1.33·31-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
| \(L(2)\) |
\(\approx\) |
\(1.757369196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.757369196\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.24T + 27T^{2} \) |
| 11 | \( 1 + 1.68T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 78.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 344.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 139.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 224.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 37.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 217.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 207.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 92.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 873.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 495.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 362.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 737.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−10.29575061742750275295200746356, −9.497451685444452198485203252771, −8.780072816160888507237623914489, −7.82613859413595161939447517165, −7.42103731088797007045301771712, −6.06714056929638308215113411565, −4.70892463157079521291711721279, −3.27245795566408279191153493515, −2.52887203887225118453106995877, −0.884583743088276070706994619284,
0.884583743088276070706994619284, 2.52887203887225118453106995877, 3.27245795566408279191153493515, 4.70892463157079521291711721279, 6.06714056929638308215113411565, 7.42103731088797007045301771712, 7.82613859413595161939447517165, 8.780072816160888507237623914489, 9.497451685444452198485203252771, 10.29575061742750275295200746356
