| L(s) = 1 | + 2·3-s + 9-s + 6·11-s + 2·13-s + 17-s + 4·19-s − 5·25-s − 4·27-s − 4·31-s + 12·33-s + 4·37-s + 4·39-s − 6·41-s + 8·43-s + 2·51-s + 6·53-s + 8·57-s − 4·61-s + 8·67-s − 2·73-s − 10·75-s − 8·79-s − 11·81-s + 6·89-s − 8·93-s − 14·97-s + 6·99-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 0.242·17-s + 0.917·19-s − 25-s − 0.769·27-s − 0.718·31-s + 2.08·33-s + 0.657·37-s + 0.640·39-s − 0.937·41-s + 1.21·43-s + 0.280·51-s + 0.824·53-s + 1.05·57-s − 0.512·61-s + 0.977·67-s − 0.234·73-s − 1.15·75-s − 0.900·79-s − 1.22·81-s + 0.635·89-s − 0.829·93-s − 1.42·97-s + 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.806066255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.806066255\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.42051328411611, −13.95806467688364, −13.65277024442186, −13.09022877692318, −12.39565839751697, −11.84097126717842, −11.47510989232572, −10.96058093409176, −10.11745164919507, −9.564692421616949, −9.259653677481715, −8.784008684701653, −8.256609321536037, −7.687882436287877, −7.160050891610368, −6.586157865565160, −5.856122894286236, −5.505994215783255, −4.413174055584165, −3.956643556846916, −3.473652067384000, −2.951880503700955, −2.049982538194369, −1.529924927062190, −0.7311590566509649,
0.7311590566509649, 1.529924927062190, 2.049982538194369, 2.951880503700955, 3.473652067384000, 3.956643556846916, 4.413174055584165, 5.505994215783255, 5.856122894286236, 6.586157865565160, 7.160050891610368, 7.687882436287877, 8.256609321536037, 8.784008684701653, 9.259653677481715, 9.564692421616949, 10.11745164919507, 10.96058093409176, 11.47510989232572, 11.84097126717842, 12.39565839751697, 13.09022877692318, 13.65277024442186, 13.95806467688364, 14.42051328411611
