| L(s) = 1 | − 3-s + 1.28·5-s − 0.442·7-s + 9-s + 0.841·11-s − 13-s − 1.28·15-s + 0.351·17-s + 3.67·19-s + 0.442·21-s + 7.04·23-s − 3.35·25-s − 27-s − 0.217·29-s − 5.14·31-s − 0.841·33-s − 0.568·35-s − 5.60·37-s + 39-s − 4.49·41-s − 10.4·43-s + 1.28·45-s − 6.90·47-s − 6.80·49-s − 0.351·51-s + 3.35·53-s + 1.08·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.574·5-s − 0.167·7-s + 0.333·9-s + 0.253·11-s − 0.277·13-s − 0.331·15-s + 0.0851·17-s + 0.843·19-s + 0.0965·21-s + 1.46·23-s − 0.670·25-s − 0.192·27-s − 0.0402·29-s − 0.924·31-s − 0.146·33-s − 0.0960·35-s − 0.922·37-s + 0.160·39-s − 0.702·41-s − 1.59·43-s + 0.191·45-s − 1.00·47-s − 0.972·49-s − 0.0491·51-s + 0.461·53-s + 0.145·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 + 0.442T + 7T^{2} \) |
| 11 | \( 1 - 0.841T + 11T^{2} \) |
| 17 | \( 1 - 0.351T + 17T^{2} \) |
| 19 | \( 1 - 3.67T + 19T^{2} \) |
| 23 | \( 1 - 7.04T + 23T^{2} \) |
| 29 | \( 1 + 0.217T + 29T^{2} \) |
| 31 | \( 1 + 5.14T + 31T^{2} \) |
| 37 | \( 1 + 5.60T + 37T^{2} \) |
| 41 | \( 1 + 4.49T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 53 | \( 1 - 3.35T + 53T^{2} \) |
| 59 | \( 1 - 4.74T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 - 7.67T + 67T^{2} \) |
| 71 | \( 1 - 7.78T + 71T^{2} \) |
| 73 | \( 1 + 4.23T + 73T^{2} \) |
| 79 | \( 1 + 0.230T + 79T^{2} \) |
| 83 | \( 1 + 7.86T + 83T^{2} \) |
| 89 | \( 1 + 5.81T + 89T^{2} \) |
| 97 | \( 1 - 9.66T + 97T^{2} \) |
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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
−7.09290186738568260881270933027, −6.70585107609904060977033179121, −5.92134213744530468862084249800, −5.18868883790016784689255960938, −4.88233440618776662214109811394, −3.69210964643690594824939977230, −3.12618206207055273948886856775, −1.98889913287750167620520545470, −1.25615845997167763041357388811, 0,
1.25615845997167763041357388811, 1.98889913287750167620520545470, 3.12618206207055273948886856775, 3.69210964643690594824939977230, 4.88233440618776662214109811394, 5.18868883790016784689255960938, 5.92134213744530468862084249800, 6.70585107609904060977033179121, 7.09290186738568260881270933027
