| L(s) = 1 | + 2·2-s − 9.22·3-s + 4·4-s − 5·5-s − 18.4·6-s + 8·8-s + 58.1·9-s − 10·10-s + 23.1·11-s − 36.9·12-s − 54.8·13-s + 46.1·15-s + 16·16-s + 52.8·17-s + 116.·18-s + 151.·19-s − 20·20-s + 46.2·22-s − 181.·23-s − 73.8·24-s + 25·25-s − 109.·26-s − 287.·27-s − 64.0·29-s + 92.2·30-s + 45.0·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.77·3-s + 0.5·4-s − 0.447·5-s − 1.25·6-s + 0.353·8-s + 2.15·9-s − 0.316·10-s + 0.633·11-s − 0.888·12-s − 1.17·13-s + 0.794·15-s + 0.250·16-s + 0.754·17-s + 1.52·18-s + 1.83·19-s − 0.223·20-s + 0.447·22-s − 1.64·23-s − 0.627·24-s + 0.200·25-s − 0.827·26-s − 2.05·27-s − 0.410·29-s + 0.561·30-s + 0.260·31-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 | \( 1 + 9.22T + 27T^{2} \) |
| 11 | \( 1 - 23.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 52.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 151.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 64.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 45.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 191.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 477.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 537.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 148.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 273.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 300.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 222.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 34.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.06e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.47e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 88.5T + 7.04e5T^{2} \) |
| 97 | \( 1 + 570.T + 9.12e5T^{2} \) |
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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
−10.20488464244320582430844206850, −9.745812871842171969442317871113, −7.83662071309337833298114634603, −7.07590600025799060356520544112, −6.16870658735517411254824024208, −5.30133405987748696126956921812, −4.61896268122217286843015170558, −3.43347264394227825771249242343, −1.45352795986866716004471793520, 0,
1.45352795986866716004471793520, 3.43347264394227825771249242343, 4.61896268122217286843015170558, 5.30133405987748696126956921812, 6.16870658735517411254824024208, 7.07590600025799060356520544112, 7.83662071309337833298114634603, 9.745812871842171969442317871113, 10.20488464244320582430844206850
