| L(s) = 1 | − 2·2-s + 3.24·3-s + 4·4-s + 5·5-s − 6.48·6-s − 8·8-s − 16.4·9-s − 10·10-s − 24.3·11-s + 12.9·12-s + 63.7·13-s + 16.2·15-s + 16·16-s − 117.·17-s + 32.9·18-s + 4.62·19-s + 20·20-s + 48.6·22-s − 164.·23-s − 25.9·24-s + 25·25-s − 127.·26-s − 141.·27-s − 124.·29-s − 32.4·30-s + 89.8·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.624·3-s + 0.5·4-s + 0.447·5-s − 0.441·6-s − 0.353·8-s − 0.610·9-s − 0.316·10-s − 0.666·11-s + 0.312·12-s + 1.35·13-s + 0.279·15-s + 0.250·16-s − 1.66·17-s + 0.431·18-s + 0.0558·19-s + 0.223·20-s + 0.471·22-s − 1.49·23-s − 0.220·24-s + 0.200·25-s − 0.961·26-s − 1.00·27-s − 0.799·29-s − 0.197·30-s + 0.520·31-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 | \( 1 - 3.24T + 27T^{2} \) |
| 11 | \( 1 + 24.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.62T + 6.85e3T^{2} \) |
| 23 | \( 1 + 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 89.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 212.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 99.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 425.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 572.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 617.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 196.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 267.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 589.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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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
−10.01005595064845591322023479731, −8.972528334284393192014314925225, −8.539849586144574296658812875702, −7.64456748925158715219529819187, −6.41090971389846369568056675257, −5.65901961234132548134986104247, −4.04970000284833102842513607455, −2.76323900378937767482222063852, −1.79451695010414984815766840004, 0,
1.79451695010414984815766840004, 2.76323900378937767482222063852, 4.04970000284833102842513607455, 5.65901961234132548134986104247, 6.41090971389846369568056675257, 7.64456748925158715219529819187, 8.539849586144574296658812875702, 8.972528334284393192014314925225, 10.01005595064845591322023479731
