| L(s) = 1 | − 2·2-s − 0.537·3-s + 4·4-s + 1.07·6-s − 8·8-s − 26.7·9-s + 14.1·11-s − 2.15·12-s − 13.9·13-s + 16·16-s + 65.7·17-s + 53.4·18-s − 95.1·19-s − 28.3·22-s − 69.4·23-s + 4.30·24-s + 27.8·26-s + 28.8·27-s + 127.·29-s + 98.3·31-s − 32·32-s − 7.62·33-s − 131.·34-s − 106.·36-s + 287.·37-s + 190.·38-s + 7.48·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.103·3-s + 0.5·4-s + 0.0731·6-s − 0.353·8-s − 0.989·9-s + 0.388·11-s − 0.0517·12-s − 0.296·13-s + 0.250·16-s + 0.938·17-s + 0.699·18-s − 1.14·19-s − 0.274·22-s − 0.629·23-s + 0.0365·24-s + 0.209·26-s + 0.205·27-s + 0.818·29-s + 0.569·31-s − 0.176·32-s − 0.0402·33-s − 0.663·34-s − 0.494·36-s + 1.27·37-s + 0.812·38-s + 0.0307·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.537T + 27T^{2} \) |
| 11 | \( 1 - 14.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 95.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 69.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 127.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 98.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 287.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 310.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 197.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 538.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 314.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 242.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 440.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 858.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 142.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 459.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 403.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 166.T + 9.12e5T^{2} \) |
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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
−8.207315037163530490963757435756, −7.72053841541096736012587403270, −6.54881524159293032510233610957, −6.13426019966041925214983223201, −5.17318813094237667250808308501, −4.13980605468645305782946541902, −3.05479561596540083159995393391, −2.24804794081790696334312736257, −1.04061061603933919575507915973, 0,
1.04061061603933919575507915973, 2.24804794081790696334312736257, 3.05479561596540083159995393391, 4.13980605468645305782946541902, 5.17318813094237667250808308501, 6.13426019966041925214983223201, 6.54881524159293032510233610957, 7.72053841541096736012587403270, 8.207315037163530490963757435756
