| L(s) = 1 | + 1.68·2-s − 3.03·3-s + 0.841·4-s − 5-s − 5.11·6-s − 1.28·7-s − 1.95·8-s + 6.20·9-s − 1.68·10-s + 5.55·11-s − 2.55·12-s − 5.66·13-s − 2.16·14-s + 3.03·15-s − 4.97·16-s − 2.71·17-s + 10.4·18-s − 7.10·19-s − 0.841·20-s + 3.89·21-s + 9.36·22-s − 0.0121·23-s + 5.92·24-s + 25-s − 9.55·26-s − 9.73·27-s − 1.08·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 1.75·3-s + 0.420·4-s − 0.447·5-s − 2.08·6-s − 0.485·7-s − 0.690·8-s + 2.06·9-s − 0.533·10-s + 1.67·11-s − 0.737·12-s − 1.57·13-s − 0.578·14-s + 0.783·15-s − 1.24·16-s − 0.657·17-s + 2.46·18-s − 1.62·19-s − 0.188·20-s + 0.850·21-s + 1.99·22-s − 0.00253·23-s + 1.20·24-s + 0.200·25-s − 1.87·26-s − 1.87·27-s − 0.204·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9905287715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9905287715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.68T + 2T^{2} \) |
| 3 | \( 1 + 3.03T + 3T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 - 5.55T + 11T^{2} \) |
| 13 | \( 1 + 5.66T + 13T^{2} \) |
| 17 | \( 1 + 2.71T + 17T^{2} \) |
| 19 | \( 1 + 7.10T + 19T^{2} \) |
| 23 | \( 1 + 0.0121T + 23T^{2} \) |
| 29 | \( 1 - 1.24T + 29T^{2} \) |
| 31 | \( 1 - 6.75T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 6.41T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 + 8.74T + 47T^{2} \) |
| 53 | \( 1 - 0.898T + 53T^{2} \) |
| 59 | \( 1 - 9.59T + 59T^{2} \) |
| 61 | \( 1 - 8.97T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 9.80T + 71T^{2} \) |
| 73 | \( 1 - 3.34T + 73T^{2} \) |
| 79 | \( 1 + 4.34T + 79T^{2} \) |
| 83 | \( 1 - 5.90T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−9.479158768649887880217475702698, −8.296658997891285401314952306813, −6.83859362822731867187991904999, −6.61481109375903461417873023523, −6.00691982493700645455668539251, −4.93122104481556630160709100402, −4.44903717751384403707829465360, −3.86495485176781260845164832129, −2.38953822514805731196436330933, −0.58600773570103304172163992416,
0.58600773570103304172163992416, 2.38953822514805731196436330933, 3.86495485176781260845164832129, 4.44903717751384403707829465360, 4.93122104481556630160709100402, 6.00691982493700645455668539251, 6.61481109375903461417873023523, 6.83859362822731867187991904999, 8.296658997891285401314952306813, 9.479158768649887880217475702698
