| L(s) = 1 | − 2.11·3-s + 0.973·7-s + 1.48·9-s + 5.38·11-s − 1.99·13-s − 2.04·17-s − 6.20·19-s − 2.05·21-s − 1.93·23-s + 3.21·27-s + 4.81·29-s + 6.64·31-s − 11.3·33-s − 0.978·37-s + 4.22·39-s + 2.73·41-s − 3.99·43-s − 7.21·47-s − 6.05·49-s + 4.32·51-s + 13.2·53-s + 13.1·57-s − 6.54·59-s + 2.72·61-s + 1.44·63-s − 9.56·67-s + 4.10·69-s + ⋯ |
| L(s) = 1 | − 1.22·3-s + 0.367·7-s + 0.493·9-s + 1.62·11-s − 0.553·13-s − 0.495·17-s − 1.42·19-s − 0.449·21-s − 0.404·23-s + 0.618·27-s + 0.894·29-s + 1.19·31-s − 1.98·33-s − 0.160·37-s + 0.676·39-s + 0.426·41-s − 0.609·43-s − 1.05·47-s − 0.864·49-s + 0.605·51-s + 1.82·53-s + 1.73·57-s − 0.851·59-s + 0.349·61-s + 0.181·63-s − 1.16·67-s + 0.493·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.11T + 3T^{2} \) |
| 7 | \( 1 - 0.973T + 7T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 + 1.99T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 + 6.20T + 19T^{2} \) |
| 23 | \( 1 + 1.93T + 23T^{2} \) |
| 29 | \( 1 - 4.81T + 29T^{2} \) |
| 31 | \( 1 - 6.64T + 31T^{2} \) |
| 37 | \( 1 + 0.978T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 + 3.99T + 43T^{2} \) |
| 47 | \( 1 + 7.21T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 6.54T + 59T^{2} \) |
| 61 | \( 1 - 2.72T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 - 5.68T + 71T^{2} \) |
| 73 | \( 1 + 9.35T + 73T^{2} \) |
| 79 | \( 1 - 3.18T + 79T^{2} \) |
| 83 | \( 1 + 6.11T + 83T^{2} \) |
| 89 | \( 1 + 3.00T + 89T^{2} \) |
| 97 | \( 1 - 5.95T + 97T^{2} \) |
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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
−6.92806658526345171329947826663, −6.57445350597975753080548488173, −6.11005875334903921226480211289, −5.28595371573511954207699346530, −4.42691411340801503830488820309, −4.25574941047859444530757241608, −3.00135576521233082360178838198, −1.96495307293593433370936327763, −1.08142014202171992926162276093, 0,
1.08142014202171992926162276093, 1.96495307293593433370936327763, 3.00135576521233082360178838198, 4.25574941047859444530757241608, 4.42691411340801503830488820309, 5.28595371573511954207699346530, 6.11005875334903921226480211289, 6.57445350597975753080548488173, 6.92806658526345171329947826663
