| L(s) = 1 | − 5.47i·3-s − 769. i·7-s + 2.15e3·9-s − 5.35e3·11-s − 1.40e4i·13-s + 1.24e4i·17-s − 5.03e3·19-s − 4.21e3·21-s − 7.51e4i·23-s − 2.37e4i·27-s − 1.95e5·29-s + 9.35e4·31-s + 2.93e4i·33-s + 1.61e5i·37-s − 7.68e4·39-s + ⋯ |
| L(s) = 1 | − 0.117i·3-s − 0.848i·7-s + 0.986·9-s − 1.21·11-s − 1.77i·13-s + 0.612i·17-s − 0.168·19-s − 0.0993·21-s − 1.28i·23-s − 0.232i·27-s − 1.48·29-s + 0.564·31-s + 0.142i·33-s + 0.524i·37-s − 0.207·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5496707001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5496707001\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5.47iT - 2.18e3T^{2} \) |
| 7 | \( 1 + 769. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 5.35e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.40e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.24e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 5.03e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.51e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.95e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 9.35e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.61e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 2.87e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.39e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 1.06e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 6.26e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 2.14e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.57e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 8.07e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.74e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 2.46e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.90e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 2.63e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.01e7iT - 8.07e13T^{2} \) |
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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
−9.999835730140858250900473426455, −8.363495010472174006771805995381, −7.76679939563189522023139780937, −6.89839613268150017995275999087, −5.70132613845018747729340362789, −4.69472066207792124140753021454, −3.62434812833550369193651627020, −2.44733903498248118252777141121, −1.06458617702928367072439025346, −0.11675091725878755086338452559,
1.58657825388125791773333991261, 2.45882792298180995653344575380, 3.85715091047571717406023538555, 4.87785857881321348783656769617, 5.79903955587755033231814099990, 7.01998418204203401866616450509, 7.72661936167938654019714960885, 9.070200175154799463988267098141, 9.505753039301857919112464781664, 10.61250598644790163595580553079
