| L(s) = 1 | − 2-s + 1.40·3-s + 4-s − 5-s − 1.40·6-s − 1.83·7-s − 8-s − 1.03·9-s + 10-s − 5.86·11-s + 1.40·12-s + 0.0643·13-s + 1.83·14-s − 1.40·15-s + 16-s − 1.12·17-s + 1.03·18-s − 0.830·19-s − 20-s − 2.57·21-s + 5.86·22-s − 3.76·23-s − 1.40·24-s + 25-s − 0.0643·26-s − 5.65·27-s − 1.83·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.809·3-s + 0.5·4-s − 0.447·5-s − 0.572·6-s − 0.694·7-s − 0.353·8-s − 0.345·9-s + 0.316·10-s − 1.76·11-s + 0.404·12-s + 0.0178·13-s + 0.491·14-s − 0.361·15-s + 0.250·16-s − 0.272·17-s + 0.244·18-s − 0.190·19-s − 0.223·20-s − 0.562·21-s + 1.25·22-s − 0.785·23-s − 0.286·24-s + 0.200·25-s − 0.0126·26-s − 1.08·27-s − 0.347·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8448989362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8448989362\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 1.40T + 3T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 11 | \( 1 + 5.86T + 11T^{2} \) |
| 13 | \( 1 - 0.0643T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 + 0.830T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 - 3.01T + 29T^{2} \) |
| 31 | \( 1 - 7.33T + 31T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 + 1.00T + 41T^{2} \) |
| 43 | \( 1 - 5.20T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 4.35T + 59T^{2} \) |
| 61 | \( 1 + 7.20T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 0.929T + 71T^{2} \) |
| 73 | \( 1 + 8.16T + 73T^{2} \) |
| 79 | \( 1 - 8.55T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 0.242T + 97T^{2} \) |
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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
−8.330785867599569865414174466641, −7.968652979857049286569739740620, −7.28821430566328307036872916442, −6.36085543105748409143435438797, −5.62792197860004689091186692275, −4.59920090220822818469338317484, −3.53483623935598577850672798053, −2.74746899948984030688561808741, −2.26260118241579810079118168183, −0.52308202516437457495441320794,
0.52308202516437457495441320794, 2.26260118241579810079118168183, 2.74746899948984030688561808741, 3.53483623935598577850672798053, 4.59920090220822818469338317484, 5.62792197860004689091186692275, 6.36085543105748409143435438797, 7.28821430566328307036872916442, 7.968652979857049286569739740620, 8.330785867599569865414174466641
