| L(s) = 1 | − 1.41·3-s − 2·5-s + 4.24·7-s − 0.999·9-s − 1.41·11-s − 4·13-s + 2.82·15-s − 17-s + 2.82·19-s − 6·21-s − 4.24·23-s − 25-s + 5.65·27-s − 6·29-s − 7.07·31-s + 2.00·33-s − 8.48·35-s − 2·37-s + 5.65·39-s − 6·41-s − 8.48·43-s + 1.99·45-s − 11.3·47-s + 10.9·49-s + 1.41·51-s − 6·53-s + 2.82·55-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 0.894·5-s + 1.60·7-s − 0.333·9-s − 0.426·11-s − 1.10·13-s + 0.730·15-s − 0.242·17-s + 0.648·19-s − 1.30·21-s − 0.884·23-s − 0.200·25-s + 1.08·27-s − 1.11·29-s − 1.27·31-s + 0.348·33-s − 1.43·35-s − 0.328·37-s + 0.905·39-s − 0.937·41-s − 1.29·43-s + 0.298·45-s − 1.65·47-s + 1.57·49-s + 0.198·51-s − 0.824·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−10.69173448155844986217917936442, −9.580240288134223750063293857238, −8.222825917361264907354840405073, −7.84941415457428037919492825974, −6.81578944938836420468302730778, −5.29279687573657963962392010010, −5.02823760011202668553500542326, −3.70696280479599982679771944031, −1.98871868663563568784051823381, 0,
1.98871868663563568784051823381, 3.70696280479599982679771944031, 5.02823760011202668553500542326, 5.29279687573657963962392010010, 6.81578944938836420468302730778, 7.84941415457428037919492825974, 8.222825917361264907354840405073, 9.580240288134223750063293857238, 10.69173448155844986217917936442
