| L(s) = 1 | + 21.5·2-s + 62.6·3-s + 338.·4-s + 1.35e3·6-s + 226.·7-s + 4.53e3·8-s + 1.74e3·9-s + 8.09e3·11-s + 2.12e4·12-s + 9.40e3·13-s + 4.88e3·14-s + 5.47e4·16-s − 2.37e3·17-s + 3.76e4·18-s − 3.34e4·19-s + 1.41e4·21-s + 1.74e5·22-s − 1.21e4·23-s + 2.84e5·24-s + 2.03e5·26-s − 2.78e4·27-s + 7.64e4·28-s − 2.37e5·29-s − 2.53e4·31-s + 6.00e5·32-s + 5.07e5·33-s − 5.11e4·34-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 1.34·3-s + 2.64·4-s + 2.55·6-s + 0.249·7-s + 3.13·8-s + 0.797·9-s + 1.83·11-s + 3.54·12-s + 1.18·13-s + 0.475·14-s + 3.34·16-s − 0.117·17-s + 1.52·18-s − 1.11·19-s + 0.333·21-s + 3.49·22-s − 0.208·23-s + 4.20·24-s + 2.26·26-s − 0.271·27-s + 0.658·28-s − 1.81·29-s − 0.152·31-s + 3.24·32-s + 2.45·33-s − 0.223·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(17.03041885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.03041885\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 21.5T + 128T^{2} \) |
| 3 | \( 1 - 62.6T + 2.18e3T^{2} \) |
| 7 | \( 1 - 226.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 8.09e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.40e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.37e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.34e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 2.37e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.53e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.29e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.78e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.98e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.19e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.02e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 9.56e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.78e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.03e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.17e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.44e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.57e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.03e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.54e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.84e5T + 8.07e13T^{2} \) |
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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
−9.480666062602978542450455129423, −8.600797090127751146982369863933, −7.64478709138129786476218816099, −6.57468671190313227744985494609, −6.01725371986181389370167598448, −4.63443769308565106784050259702, −3.71126006027200858333247785959, −3.47903165202637583345538710259, −2.08652606702200580039958804493, −1.53246890199621523474497085486,
1.53246890199621523474497085486, 2.08652606702200580039958804493, 3.47903165202637583345538710259, 3.71126006027200858333247785959, 4.63443769308565106784050259702, 6.01725371986181389370167598448, 6.57468671190313227744985494609, 7.64478709138129786476218816099, 8.600797090127751146982369863933, 9.480666062602978542450455129423
