| L(s) = 1 | + 437.·5-s − 1.72e3·7-s + 4.22e3·11-s + 6.63e3·13-s − 3.41e4·17-s − 2.03e4·19-s + 4.59e4·23-s + 1.13e5·25-s + 5.14e4·29-s + 1.51e4·31-s − 7.53e5·35-s + 3.84e5·37-s − 6.02e5·41-s − 2.48e5·43-s − 7.18e5·47-s + 2.14e6·49-s + 1.39e6·53-s + 1.84e6·55-s − 1.19e6·59-s − 7.64e5·61-s + 2.90e6·65-s − 2.54e6·67-s − 4.66e6·71-s − 3.39e5·73-s − 7.27e6·77-s − 3.28e5·79-s − 1.77e6·83-s + ⋯ |
| L(s) = 1 | + 1.56·5-s − 1.89·7-s + 0.957·11-s + 0.838·13-s − 1.68·17-s − 0.680·19-s + 0.788·23-s + 1.45·25-s + 0.391·29-s + 0.0915·31-s − 2.97·35-s + 1.24·37-s − 1.36·41-s − 0.476·43-s − 1.01·47-s + 2.60·49-s + 1.28·53-s + 1.49·55-s − 0.758·59-s − 0.431·61-s + 1.31·65-s − 1.03·67-s − 1.54·71-s − 0.102·73-s − 1.81·77-s − 0.0749·79-s − 0.340·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 437.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.72e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.22e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.63e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.41e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.03e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.59e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.14e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.51e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.84e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.02e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.48e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.18e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.39e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.19e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 7.64e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.54e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.66e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.39e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.28e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.82e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.28e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.100073887170692865605663019784, −8.822796298767849324890069573207, −6.82531833561393280607771999530, −6.44589230149212941438341144889, −5.89122361169047238826356706724, −4.46643057118904091196785495970, −3.30877824104977025324415868624, −2.38226054947486821669973285801, −1.29512741379349892952575180090, 0,
1.29512741379349892952575180090, 2.38226054947486821669973285801, 3.30877824104977025324415868624, 4.46643057118904091196785495970, 5.89122361169047238826356706724, 6.44589230149212941438341144889, 6.82531833561393280607771999530, 8.822796298767849324890069573207, 9.100073887170692865605663019784
