| L(s) = 1 | − 2-s − 3-s + 4-s + 1.46·5-s + 6-s − 2.29·7-s − 8-s + 9-s − 1.46·10-s − 1.24·11-s − 12-s − 6.90·13-s + 2.29·14-s − 1.46·15-s + 16-s + 5.12·17-s − 18-s − 3.85·19-s + 1.46·20-s + 2.29·21-s + 1.24·22-s + 23-s + 24-s − 2.85·25-s + 6.90·26-s − 27-s − 2.29·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.655·5-s + 0.408·6-s − 0.867·7-s − 0.353·8-s + 0.333·9-s − 0.463·10-s − 0.375·11-s − 0.288·12-s − 1.91·13-s + 0.613·14-s − 0.378·15-s + 0.250·16-s + 1.24·17-s − 0.235·18-s − 0.883·19-s + 0.327·20-s + 0.500·21-s + 0.265·22-s + 0.208·23-s + 0.204·24-s − 0.570·25-s + 1.35·26-s − 0.192·27-s − 0.433·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6657857858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6657857858\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 1.46T + 5T^{2} \) |
| 7 | \( 1 + 2.29T + 7T^{2} \) |
| 11 | \( 1 + 1.24T + 11T^{2} \) |
| 13 | \( 1 + 6.90T + 13T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 + 3.85T + 19T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 3.53T + 41T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 + 1.27T + 53T^{2} \) |
| 59 | \( 1 + 7.36T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 6.17T + 79T^{2} \) |
| 83 | \( 1 - 1.61T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 2.68T + 97T^{2} \) |
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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
−8.467614755818057075326147825561, −7.62762383713886480494401875781, −6.99913235269792924328859452546, −6.36207747869269921871627245759, −5.51874732136206612015158149986, −5.00066059664194345080164445834, −3.73190445303366159665508059577, −2.68426144999371300567791138423, −1.94125957294907385844774291203, −0.51000000635580593785659963192,
0.51000000635580593785659963192, 1.94125957294907385844774291203, 2.68426144999371300567791138423, 3.73190445303366159665508059577, 5.00066059664194345080164445834, 5.51874732136206612015158149986, 6.36207747869269921871627245759, 6.99913235269792924328859452546, 7.62762383713886480494401875781, 8.467614755818057075326147825561
