L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 7.47·5-s − 216·6-s + 343·7-s − 512·8-s + 729·9-s − 59.7·10-s + 5.39e3·11-s + 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s + 201.·15-s + 4.09e3·16-s + 2.38e4·17-s − 5.83e3·18-s + 4.68e4·19-s + 478.·20-s + 9.26e3·21-s − 4.31e4·22-s + 7.39e4·23-s − 1.38e4·24-s − 7.80e4·25-s + 1.75e4·26-s + 1.96e4·27-s + 2.19e4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.0267·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.0189·10-s + 1.22·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.0154·15-s + 0.250·16-s + 1.17·17-s − 0.235·18-s + 1.56·19-s + 0.0133·20-s + 0.218·21-s − 0.863·22-s + 1.26·23-s − 0.204·24-s − 0.999·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.885345391\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.885345391\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 + 2.19e3T \) |
good | 5 | \( 1 - 7.47T + 7.81e4T^{2} \) |
| 11 | \( 1 - 5.39e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.38e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.68e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.39e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.75e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.77e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.41e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.87e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.42e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.54e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.33e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.35e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.13e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.21e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.08e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.07e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.87e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.58e6T + 8.07e13T^{2} \) |
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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
−9.548793913623993083477658690875, −8.928697649034410630464277882725, −7.83482855226227701725855853560, −7.34433755240372040709862950850, −6.21019029283432869723791594542, −5.08590115815536256399360637679, −3.76959889395914711682302460423, −2.86997251898482629739027851676, −1.56089353411444017873469008470, −0.878120644194415075509068911265,
0.878120644194415075509068911265, 1.56089353411444017873469008470, 2.86997251898482629739027851676, 3.76959889395914711682302460423, 5.08590115815536256399360637679, 6.21019029283432869723791594542, 7.34433755240372040709862950850, 7.83482855226227701725855853560, 8.928697649034410630464277882725, 9.548793913623993083477658690875