| L(s) = 1 | + 373.·5-s + 343·7-s + 4.60e3·11-s − 5.28e3·13-s + 1.60e4·17-s + 2.83e4·19-s + 3.90e4·23-s + 6.14e4·25-s + 9.73e4·29-s − 2.41e5·31-s + 1.28e5·35-s + 2.49e5·37-s + 1.08e5·41-s − 2.28e5·43-s − 3.52e5·47-s + 1.17e5·49-s − 7.78e5·53-s + 1.72e6·55-s − 1.76e6·59-s + 2.60e5·61-s − 1.97e6·65-s − 1.24e6·67-s + 1.36e6·71-s + 1.75e6·73-s + 1.57e6·77-s + 6.81e6·79-s + 7.92e6·83-s + ⋯ |
| L(s) = 1 | + 1.33·5-s + 0.377·7-s + 1.04·11-s − 0.667·13-s + 0.793·17-s + 0.949·19-s + 0.668·23-s + 0.786·25-s + 0.741·29-s − 1.45·31-s + 0.505·35-s + 0.808·37-s + 0.245·41-s − 0.438·43-s − 0.495·47-s + 0.142·49-s − 0.718·53-s + 1.39·55-s − 1.12·59-s + 0.146·61-s − 0.892·65-s − 0.504·67-s + 0.453·71-s + 0.527·73-s + 0.394·77-s + 1.55·79-s + 1.52·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.439611097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.439611097\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 5 | \( 1 - 373.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.60e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.28e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.60e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.83e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.90e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.73e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.41e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.49e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.08e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.52e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.78e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.76e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.60e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.24e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.36e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.75e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.81e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.92e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.49e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.93e5T + 8.07e13T^{2} \) |
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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
−10.70666662874727957510885609623, −9.624094750852025448855275546814, −9.229853571291429333450961154620, −7.81583727520378254295417038126, −6.72024297712191847758432074450, −5.71648795586946143076185612135, −4.82477475196548627588423593600, −3.27918655609939795175772085419, −1.96497674369795894388626784417, −1.01785267369795611692021190370,
1.01785267369795611692021190370, 1.96497674369795894388626784417, 3.27918655609939795175772085419, 4.82477475196548627588423593600, 5.71648795586946143076185612135, 6.72024297712191847758432074450, 7.81583727520378254295417038126, 9.229853571291429333450961154620, 9.624094750852025448855275546814, 10.70666662874727957510885609623
