| L(s) = 1 | − 20.4·2-s − 94.4·4-s + 2.58e3·5-s − 1.76e3·7-s + 1.23e4·8-s − 5.27e4·10-s + 7.45e4·11-s + 7.10e3·13-s + 3.61e4·14-s − 2.04e5·16-s + 2.68e5·17-s + 5.72e5·19-s − 2.44e5·20-s − 1.52e6·22-s − 1.28e6·23-s + 4.72e6·25-s − 1.45e5·26-s + 1.67e5·28-s + 6.39e6·29-s − 2.00e6·31-s − 2.15e6·32-s − 5.48e6·34-s − 4.57e6·35-s + 1.49e7·37-s − 1.16e7·38-s + 3.20e7·40-s + 1.77e7·41-s + ⋯ |
| L(s) = 1 | − 0.903·2-s − 0.184·4-s + 1.84·5-s − 0.278·7-s + 1.06·8-s − 1.66·10-s + 1.53·11-s + 0.0689·13-s + 0.251·14-s − 0.781·16-s + 0.780·17-s + 1.00·19-s − 0.341·20-s − 1.38·22-s − 0.960·23-s + 2.41·25-s − 0.0622·26-s + 0.0514·28-s + 1.68·29-s − 0.389·31-s − 0.364·32-s − 0.704·34-s − 0.515·35-s + 1.30·37-s − 0.909·38-s + 1.97·40-s + 0.980·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.593240816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.593240816\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 3.41e6T \) |
| good | 2 | \( 1 + 20.4T + 512T^{2} \) |
| 5 | \( 1 - 2.58e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.76e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.45e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 7.10e3T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.72e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.28e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.39e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.00e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.49e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.77e7T + 3.27e14T^{2} \) |
| 47 | \( 1 + 1.13e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.02e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 8.07e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.21e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.55e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.87e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.82e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.54e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.00e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.13e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.37e8T + 7.60e17T^{2} \) |
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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
−9.823678021635291273577933821451, −9.160559824046014791101485025429, −8.280031087353686860455168211409, −6.97853000574674251183638932157, −6.14747457194021736410326249440, −5.25233172014662463837689932610, −3.99138565118640479050277181467, −2.54635095210160319193624796999, −1.37767814503950966967267946769, −0.925214381891312157364795752195,
0.925214381891312157364795752195, 1.37767814503950966967267946769, 2.54635095210160319193624796999, 3.99138565118640479050277181467, 5.25233172014662463837689932610, 6.14747457194021736410326249440, 6.97853000574674251183638932157, 8.280031087353686860455168211409, 9.160559824046014791101485025429, 9.823678021635291273577933821451
