| L(s) = 1 | − 2·2-s − 5.78·3-s + 4·4-s − 5·5-s + 11.5·6-s − 8·8-s + 6.43·9-s + 10·10-s − 20.4·11-s − 23.1·12-s + 53.1·13-s + 28.9·15-s + 16·16-s − 27.7·17-s − 12.8·18-s + 143.·19-s − 20·20-s + 40.8·22-s − 200.·23-s + 46.2·24-s + 25·25-s − 106.·26-s + 118.·27-s + 113.·29-s − 57.8·30-s + 105.·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.11·3-s + 0.5·4-s − 0.447·5-s + 0.786·6-s − 0.353·8-s + 0.238·9-s + 0.316·10-s − 0.560·11-s − 0.556·12-s + 1.13·13-s + 0.497·15-s + 0.250·16-s − 0.395·17-s − 0.168·18-s + 1.72·19-s − 0.223·20-s + 0.396·22-s − 1.81·23-s + 0.393·24-s + 0.200·25-s − 0.801·26-s + 0.847·27-s + 0.724·29-s − 0.351·30-s + 0.611·31-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5.78T + 27T^{2} \) |
| 11 | \( 1 + 20.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 200.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 113.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 105.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 1.38T + 5.06e4T^{2} \) |
| 41 | \( 1 - 226.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 268.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 27.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 74.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 665.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 509.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 981.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 144.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 735.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 146.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 712.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 606.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 771.T + 9.12e5T^{2} \) |
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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
−10.34776788013057545236386032035, −9.257107474720428561982165086115, −8.218072605488178108090091433399, −7.47298215952957045797716804917, −6.26890458702139957394183513160, −5.67672375789941247705982286114, −4.39236955133977824638796963731, −2.96210175062548845152882749686, −1.18544078469758491623642159273, 0,
1.18544078469758491623642159273, 2.96210175062548845152882749686, 4.39236955133977824638796963731, 5.67672375789941247705982286114, 6.26890458702139957394183513160, 7.47298215952957045797716804917, 8.218072605488178108090091433399, 9.257107474720428561982165086115, 10.34776788013057545236386032035
