| L(s) = 1 | − 9.23·2-s − 79.3·3-s − 42.7·4-s + 732.·6-s + 343·7-s + 1.57e3·8-s + 4.10e3·9-s + 7.56e3·11-s + 3.38e3·12-s + 6.57e3·13-s − 3.16e3·14-s − 9.09e3·16-s − 3.85e4·17-s − 3.79e4·18-s + 2.78e4·19-s − 2.72e4·21-s − 6.98e4·22-s + 5.65e4·23-s − 1.25e5·24-s − 6.06e4·26-s − 1.52e5·27-s − 1.46e4·28-s + 2.13e5·29-s + 1.89e4·31-s − 1.17e5·32-s − 5.99e5·33-s + 3.55e5·34-s + ⋯ |
| L(s) = 1 | − 0.816·2-s − 1.69·3-s − 0.333·4-s + 1.38·6-s + 0.377·7-s + 1.08·8-s + 1.87·9-s + 1.71·11-s + 0.565·12-s + 0.829·13-s − 0.308·14-s − 0.554·16-s − 1.90·17-s − 1.53·18-s + 0.931·19-s − 0.641·21-s − 1.39·22-s + 0.968·23-s − 1.84·24-s − 0.677·26-s − 1.48·27-s − 0.126·28-s + 1.62·29-s + 0.114·31-s − 0.635·32-s − 2.90·33-s + 1.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7714029059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7714029059\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 + 9.23T + 128T^{2} \) |
| 3 | \( 1 + 79.3T + 2.18e3T^{2} \) |
| 11 | \( 1 - 7.56e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.57e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.85e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.78e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.65e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.13e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.89e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.58e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.71e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.93e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.52e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.21e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.62e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 9.81e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.20e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.27e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.51e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.45e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.84e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.49e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.88e6T + 8.07e13T^{2} \) |
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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
−11.24803035257121369597036949270, −10.61445727253141473308566881646, −9.397919295847072720286192810952, −8.611889224770389209145902184318, −7.02982788279353945595705935763, −6.32127626021230556747373066213, −4.94179110198879298500687246206, −4.14311337431917602156328174821, −1.39781645709569522238607197875, −0.69812081620072391098736109676,
0.69812081620072391098736109676, 1.39781645709569522238607197875, 4.14311337431917602156328174821, 4.94179110198879298500687246206, 6.32127626021230556747373066213, 7.02982788279353945595705935763, 8.611889224770389209145902184318, 9.397919295847072720286192810952, 10.61445727253141473308566881646, 11.24803035257121369597036949270
