| L(s) = 1 | + 2-s + 0.803·3-s + 4-s + 0.712·5-s + 0.803·6-s + 8-s − 2.35·9-s + 0.712·10-s + 2.61·11-s + 0.803·12-s + 1.95·13-s + 0.572·15-s + 16-s − 3.59·17-s − 2.35·18-s + 7.06·19-s + 0.712·20-s + 2.61·22-s + 3.24·23-s + 0.803·24-s − 4.49·25-s + 1.95·26-s − 4.30·27-s + 29-s + 0.572·30-s + 8.32·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.464·3-s + 0.5·4-s + 0.318·5-s + 0.328·6-s + 0.353·8-s − 0.784·9-s + 0.225·10-s + 0.789·11-s + 0.232·12-s + 0.542·13-s + 0.147·15-s + 0.250·16-s − 0.870·17-s − 0.554·18-s + 1.62·19-s + 0.159·20-s + 0.558·22-s + 0.677·23-s + 0.164·24-s − 0.898·25-s + 0.383·26-s − 0.828·27-s + 0.185·29-s + 0.104·30-s + 1.49·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.804542356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.804542356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.803T + 3T^{2} \) |
| 5 | \( 1 - 0.712T + 5T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 - 1.95T + 13T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 - 7.06T + 19T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 31 | \( 1 - 8.32T + 31T^{2} \) |
| 37 | \( 1 - 0.815T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 3.35T + 43T^{2} \) |
| 47 | \( 1 - 0.930T + 47T^{2} \) |
| 53 | \( 1 + 5.55T + 53T^{2} \) |
| 59 | \( 1 - 8.89T + 59T^{2} \) |
| 61 | \( 1 - 1.48T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 - 6.03T + 71T^{2} \) |
| 73 | \( 1 + 7.83T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 3.96T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.842976961498095975266564426452, −8.004572904581858656020050659828, −7.21128404239638558041662869193, −6.30560354017669030910542969573, −5.79082006663177436679740641525, −4.86180209943083427943499332566, −3.95630911364223731040052310524, −3.13854111570879360629075072452, −2.37081180778189140234174754962, −1.13839530975477412920278637312,
1.13839530975477412920278637312, 2.37081180778189140234174754962, 3.13854111570879360629075072452, 3.95630911364223731040052310524, 4.86180209943083427943499332566, 5.79082006663177436679740641525, 6.30560354017669030910542969573, 7.21128404239638558041662869193, 8.004572904581858656020050659828, 8.842976961498095975266564426452
