L(s) = 1 | + 9.68·2-s + 60.1·3-s − 34.2·4-s − 158.·5-s + 582.·6-s − 343·7-s − 1.57e3·8-s + 1.42e3·9-s − 1.53e3·10-s − 4.07e3·11-s − 2.05e3·12-s + 2.19e3·13-s − 3.32e3·14-s − 9.53e3·15-s − 1.08e4·16-s + 945.·17-s + 1.38e4·18-s − 5.59e4·19-s + 5.43e3·20-s − 2.06e4·21-s − 3.94e4·22-s + 4.47e4·23-s − 9.44e4·24-s − 5.29e4·25-s + 2.12e4·26-s − 4.56e4·27-s + 1.17e4·28-s + ⋯ |
L(s) = 1 | + 0.855·2-s + 1.28·3-s − 0.267·4-s − 0.567·5-s + 1.10·6-s − 0.377·7-s − 1.08·8-s + 0.653·9-s − 0.485·10-s − 0.922·11-s − 0.344·12-s + 0.277·13-s − 0.323·14-s − 0.729·15-s − 0.660·16-s + 0.0466·17-s + 0.559·18-s − 1.87·19-s + 0.151·20-s − 0.485·21-s − 0.789·22-s + 0.766·23-s − 1.39·24-s − 0.678·25-s + 0.237·26-s − 0.445·27-s + 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + 343T \) |
| 13 | \( 1 - 2.19e3T \) |
good | 2 | \( 1 - 9.68T + 128T^{2} \) |
| 3 | \( 1 - 60.1T + 2.18e3T^{2} \) |
| 5 | \( 1 + 158.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 4.07e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 945.T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.59e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.47e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.95e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.50e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.05e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.01e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.40e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.40e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.59e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.57e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.45e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.63e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.81e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.55e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.77e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.87e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.57e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.29e6T + 8.07e13T^{2} \) |
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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
−12.75749603060118220547675083591, −11.22263942557571896305718772664, −9.716069657711537469789533464221, −8.649998793380084238639919626995, −7.84597185099194797704947190090, −6.16405428041673999791054797939, −4.56170797026164947582303115591, −3.52663397569566864305344730612, −2.49758646553294322302156472477, 0,
2.49758646553294322302156472477, 3.52663397569566864305344730612, 4.56170797026164947582303115591, 6.16405428041673999791054797939, 7.84597185099194797704947190090, 8.649998793380084238639919626995, 9.716069657711537469789533464221, 11.22263942557571896305718772664, 12.75749603060118220547675083591