| L(s) = 1 | + 15.1·2-s − 27·3-s + 101.·4-s − 408.·6-s + 198.·7-s − 407.·8-s + 729·9-s − 5.26e3·11-s − 2.72e3·12-s + 1.21e3·13-s + 3.01e3·14-s − 1.91e4·16-s − 3.45e4·17-s + 1.10e4·18-s + 1.86e4·19-s − 5.37e3·21-s − 7.97e4·22-s − 3.33e4·23-s + 1.09e4·24-s + 1.83e4·26-s − 1.96e4·27-s + 2.01e4·28-s − 1.78e5·29-s − 2.37e5·31-s − 2.37e5·32-s + 1.42e5·33-s − 5.23e5·34-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 0.577·3-s + 0.789·4-s − 0.772·6-s + 0.219·7-s − 0.281·8-s + 0.333·9-s − 1.19·11-s − 0.455·12-s + 0.153·13-s + 0.293·14-s − 1.16·16-s − 1.70·17-s + 0.445·18-s + 0.622·19-s − 0.126·21-s − 1.59·22-s − 0.571·23-s + 0.162·24-s + 0.204·26-s − 0.192·27-s + 0.173·28-s − 1.35·29-s − 1.43·31-s − 1.27·32-s + 0.688·33-s − 2.28·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 15.1T + 128T^{2} \) |
| 7 | \( 1 - 198.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.26e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.21e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.86e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.33e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.78e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.37e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.82e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.93e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.81e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.66e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.75e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.15e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.29e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.71e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.67e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.44e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.71e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.52e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.44e7T + 8.07e13T^{2} \) |
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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
−12.90706714213168671220048834459, −11.58679240003902503944625915739, −10.82398959412116427748702670180, −9.197262061340172004388268200670, −7.49630344925261977910404841262, −6.07826075964250517172888801641, −5.13896581388417203369936464353, −4.01831613773039630036042194184, −2.35654417944878452968724239249, 0,
2.35654417944878452968724239249, 4.01831613773039630036042194184, 5.13896581388417203369936464353, 6.07826075964250517172888801641, 7.49630344925261977910404841262, 9.197262061340172004388268200670, 10.82398959412116427748702670180, 11.58679240003902503944625915739, 12.90706714213168671220048834459
