| L(s) = 1 | − 2-s − 3-s + 4-s − 3.65·5-s + 6-s − 2.98·7-s − 8-s + 9-s + 3.65·10-s + 3.60·11-s − 12-s + 13-s + 2.98·14-s + 3.65·15-s + 16-s − 3.22·17-s − 18-s + 3.44·19-s − 3.65·20-s + 2.98·21-s − 3.60·22-s + 3.62·23-s + 24-s + 8.35·25-s − 26-s − 27-s − 2.98·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.63·5-s + 0.408·6-s − 1.12·7-s − 0.353·8-s + 0.333·9-s + 1.15·10-s + 1.08·11-s − 0.288·12-s + 0.277·13-s + 0.797·14-s + 0.943·15-s + 0.250·16-s − 0.781·17-s − 0.235·18-s + 0.791·19-s − 0.817·20-s + 0.651·21-s − 0.769·22-s + 0.755·23-s + 0.204·24-s + 1.67·25-s − 0.196·26-s − 0.192·27-s − 0.564·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6018093083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6018093083\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 3.65T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 19 | \( 1 - 3.44T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 1.67T + 37T^{2} \) |
| 41 | \( 1 + 8.55T + 41T^{2} \) |
| 43 | \( 1 - 5.42T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 1.05T + 53T^{2} \) |
| 59 | \( 1 + 5.39T + 59T^{2} \) |
| 61 | \( 1 + 6.09T + 61T^{2} \) |
| 67 | \( 1 - 0.180T + 67T^{2} \) |
| 71 | \( 1 + 2.75T + 71T^{2} \) |
| 73 | \( 1 + 6.96T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 4.83T + 97T^{2} \) |
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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
−7.81718179373297955044422998894, −7.08638518783397380194194369710, −6.52989529474190563137172219710, −6.22221730289671379057315043878, −4.80992930383466473792396908168, −4.33864933353556614375512149727, −3.33843465109529537742336796587, −2.95145343245712398273314888586, −1.27939950253271391610590256979, −0.49925470795214414458870578708,
0.49925470795214414458870578708, 1.27939950253271391610590256979, 2.95145343245712398273314888586, 3.33843465109529537742336796587, 4.33864933353556614375512149727, 4.80992930383466473792396908168, 6.22221730289671379057315043878, 6.52989529474190563137172219710, 7.08638518783397380194194369710, 7.81718179373297955044422998894
