| L(s) = 1 | − 2-s + 2.60·3-s + 4-s + 5-s − 2.60·6-s − 0.833·7-s − 8-s + 3.79·9-s − 10-s − 3.84·11-s + 2.60·12-s + 0.833·14-s + 2.60·15-s + 16-s + 8.04·17-s − 3.79·18-s + 3.04·19-s + 20-s − 2.17·21-s + 3.84·22-s + 4.17·23-s − 2.60·24-s + 25-s + 2.08·27-s − 0.833·28-s + 8.57·29-s − 2.60·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.50·3-s + 0.5·4-s + 0.447·5-s − 1.06·6-s − 0.315·7-s − 0.353·8-s + 1.26·9-s − 0.316·10-s − 1.15·11-s + 0.752·12-s + 0.222·14-s + 0.673·15-s + 0.250·16-s + 1.95·17-s − 0.895·18-s + 0.697·19-s + 0.223·20-s − 0.474·21-s + 0.818·22-s + 0.870·23-s − 0.532·24-s + 0.200·25-s + 0.400·27-s − 0.157·28-s + 1.59·29-s − 0.476·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.969533122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.969533122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 7 | \( 1 + 0.833T + 7T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 8.04T + 17T^{2} \) |
| 19 | \( 1 - 3.04T + 19T^{2} \) |
| 23 | \( 1 - 4.17T + 23T^{2} \) |
| 29 | \( 1 - 8.57T + 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 - 5.40T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8.69T + 43T^{2} \) |
| 47 | \( 1 + 9.97T + 47T^{2} \) |
| 53 | \( 1 - 1.05T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 0.332T + 67T^{2} \) |
| 71 | \( 1 - 2.78T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 83 | \( 1 + 5.54T + 83T^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 + 5.75T + 97T^{2} \) |
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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
−9.798173532131179451047055153833, −9.649128609644137113340242255626, −8.463411836512042282188083461444, −7.963632684991090511785001729045, −7.25490321242080760393992491131, −6.03735093422304091538417842084, −4.88072838007739738335612338638, −3.10823022883554242967015643899, −2.88459958488978761581862010164, −1.36286724370304758495087797159,
1.36286724370304758495087797159, 2.88459958488978761581862010164, 3.10823022883554242967015643899, 4.88072838007739738335612338638, 6.03735093422304091538417842084, 7.25490321242080760393992491131, 7.963632684991090511785001729045, 8.463411836512042282188083461444, 9.649128609644137113340242255626, 9.798173532131179451047055153833
