| L(s) = 1 | + 2.22·3-s − 3.22·5-s + 0.00440·7-s + 1.95·9-s + 1.22·11-s + 4.82·13-s − 7.18·15-s − 2.43·17-s − 7.24·19-s + 0.00980·21-s − 4.27·23-s + 5.40·25-s − 2.32·27-s − 2.39·29-s + 5.78·31-s + 2.71·33-s − 0.0142·35-s + 2.84·37-s + 10.7·39-s + 0.665·41-s + 2.23·43-s − 6.30·45-s − 6.13·47-s − 6.99·49-s − 5.42·51-s − 13.9·53-s − 3.93·55-s + ⋯ |
| L(s) = 1 | + 1.28·3-s − 1.44·5-s + 0.00166·7-s + 0.651·9-s + 0.367·11-s + 1.33·13-s − 1.85·15-s − 0.591·17-s − 1.66·19-s + 0.00213·21-s − 0.890·23-s + 1.08·25-s − 0.448·27-s − 0.444·29-s + 1.03·31-s + 0.472·33-s − 0.00240·35-s + 0.468·37-s + 1.71·39-s + 0.103·41-s + 0.341·43-s − 0.939·45-s − 0.894·47-s − 0.999·49-s − 0.759·51-s − 1.91·53-s − 0.530·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 + 3.22T + 5T^{2} \) |
| 7 | \( 1 - 0.00440T + 7T^{2} \) |
| 11 | \( 1 - 1.22T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 2.43T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 + 2.39T + 29T^{2} \) |
| 31 | \( 1 - 5.78T + 31T^{2} \) |
| 37 | \( 1 - 2.84T + 37T^{2} \) |
| 41 | \( 1 - 0.665T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 6.13T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 + 8.00T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 8.09T + 67T^{2} \) |
| 71 | \( 1 + 2.79T + 71T^{2} \) |
| 73 | \( 1 - 1.66T + 73T^{2} \) |
| 79 | \( 1 - 3.84T + 79T^{2} \) |
| 83 | \( 1 - 8.07T + 83T^{2} \) |
| 89 | \( 1 + 5.05T + 89T^{2} \) |
| 97 | \( 1 + 4.28T + 97T^{2} \) |
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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
−8.104394487627768967174430349757, −7.77551128499400989107768086023, −6.63475190506764380152190458029, −6.15575428040868205582011784216, −4.64475488522018143734999136715, −3.99594205346482175836005017779, −3.57010502683064299117806871064, −2.63256459479996758622338700795, −1.58598519656033741497375630556, 0,
1.58598519656033741497375630556, 2.63256459479996758622338700795, 3.57010502683064299117806871064, 3.99594205346482175836005017779, 4.64475488522018143734999136715, 6.15575428040868205582011784216, 6.63475190506764380152190458029, 7.77551128499400989107768086023, 8.104394487627768967174430349757
