| L(s) = 1 | + 16.5·2-s − 27·3-s + 147.·4-s + 174.·5-s − 447.·6-s + 1.11e3·7-s + 319.·8-s + 729·9-s + 2.90e3·10-s + 3.45e3·11-s − 3.97e3·12-s − 368.·13-s + 1.84e4·14-s − 4.72e3·15-s − 1.35e4·16-s + 1.20e4·17-s + 1.20e4·18-s − 2.03e4·19-s + 2.57e4·20-s − 2.99e4·21-s + 5.73e4·22-s + 2.76e4·23-s − 8.62e3·24-s − 4.75e4·25-s − 6.11e3·26-s − 1.96e4·27-s + 1.63e5·28-s + ⋯ |
| L(s) = 1 | + 1.46·2-s − 0.577·3-s + 1.15·4-s + 0.626·5-s − 0.846·6-s + 1.22·7-s + 0.220·8-s + 0.333·9-s + 0.918·10-s + 0.783·11-s − 0.664·12-s − 0.0465·13-s + 1.79·14-s − 0.361·15-s − 0.827·16-s + 0.593·17-s + 0.488·18-s − 0.682·19-s + 0.720·20-s − 0.706·21-s + 1.14·22-s + 0.474·23-s − 0.127·24-s − 0.608·25-s − 0.0682·26-s − 0.192·27-s + 1.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.377793233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.377793233\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 16.5T + 128T^{2} \) |
| 5 | \( 1 - 174.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.11e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.45e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 368.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.20e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.03e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.76e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.24e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.86e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.41e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 5.22e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.89e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.16e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.37e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.64e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.91e6T + 8.07e13T^{2} \) |
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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
−11.68402711840782443280612679514, −10.83992658316770169780812356929, −9.556279385537436520401952043710, −8.157330379075469906843986201609, −6.67844369929411066913512285429, −5.85557290746436087734334969153, −4.89224545348564307056459316963, −4.10340674757939942966180505212, −2.47764219970074304637001463125, −1.16199140779909799718345720033,
1.16199140779909799718345720033, 2.47764219970074304637001463125, 4.10340674757939942966180505212, 4.89224545348564307056459316963, 5.85557290746436087734334969153, 6.67844369929411066913512285429, 8.157330379075469906843986201609, 9.556279385537436520401952043710, 10.83992658316770169780812356929, 11.68402711840782443280612679514
