| L(s) = 1 | + 3-s − 2·4-s − 4·7-s + 9-s + 3·11-s − 2·12-s + 13-s + 4·16-s − 19-s − 4·21-s + 9·23-s + 27-s + 8·28-s − 6·29-s − 2·31-s + 3·33-s − 2·36-s − 4·37-s + 39-s + 3·41-s + 7·43-s − 6·44-s + 6·47-s + 4·48-s + 9·49-s − 2·52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 1.51·7-s + 1/3·9-s + 0.904·11-s − 0.577·12-s + 0.277·13-s + 16-s − 0.229·19-s − 0.872·21-s + 1.87·23-s + 0.192·27-s + 1.51·28-s − 1.11·29-s − 0.359·31-s + 0.522·33-s − 1/3·36-s − 0.657·37-s + 0.160·39-s + 0.468·41-s + 1.06·43-s − 0.904·44-s + 0.875·47-s + 0.577·48-s + 9/7·49-s − 0.277·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.679156965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.679156965\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.39320183500682, −14.94800206193470, −14.51257127824168, −13.76891825788667, −13.41853074855706, −12.93664373845166, −12.50814305261080, −11.97189225367663, −10.96361283417758, −10.56856349050627, −9.751169147108564, −9.287723972042963, −9.056059189209078, −8.569700646198540, −7.630754512995864, −7.081995823211992, −6.516097403696149, −5.782008241288634, −5.202896508746613, −4.221724060819919, −3.819059514505526, −3.252575997364471, −2.569403306836858, −1.375575207823029, −0.5544980049861026,
0.5544980049861026, 1.375575207823029, 2.569403306836858, 3.252575997364471, 3.819059514505526, 4.221724060819919, 5.202896508746613, 5.782008241288634, 6.516097403696149, 7.081995823211992, 7.630754512995864, 8.569700646198540, 9.056059189209078, 9.287723972042963, 9.751169147108564, 10.56856349050627, 10.96361283417758, 11.97189225367663, 12.50814305261080, 12.93664373845166, 13.41853074855706, 13.76891825788667, 14.51257127824168, 14.94800206193470, 15.39320183500682
