| L(s) = 1 | − 45.2·2-s + 1.34e3i·3-s + 2.04e3·4-s + 1.64e4i·5-s − 6.06e4i·6-s − 9.26e4·8-s − 1.26e6·9-s − 7.43e5i·10-s + 9.28e5·11-s + 2.74e6i·12-s + 8.24e6i·13-s − 2.20e7·15-s + 4.19e6·16-s − 1.35e7i·17-s + 5.73e7·18-s + 4.48e7i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.83i·3-s + 0.500·4-s + 1.05i·5-s − 1.30i·6-s − 0.353·8-s − 2.38·9-s − 0.743i·10-s + 0.523·11-s + 0.919i·12-s + 1.70i·13-s − 1.93·15-s + 0.250·16-s − 0.561i·17-s + 1.68·18-s + 0.954i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.3558487720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3558487720\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 45.2T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.34e3iT - 5.31e5T^{2} \) |
| 5 | \( 1 - 1.64e4iT - 2.44e8T^{2} \) |
| 11 | \( 1 - 9.28e5T + 3.13e12T^{2} \) |
| 13 | \( 1 - 8.24e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + 1.35e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 - 4.48e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 + 7.07e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + 5.34e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + 1.32e9iT - 7.87e17T^{2} \) |
| 37 | \( 1 + 1.05e9T + 6.58e18T^{2} \) |
| 41 | \( 1 - 1.78e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 4.20e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + 1.73e10iT - 1.16e20T^{2} \) |
| 53 | \( 1 + 4.93e9T + 4.91e20T^{2} \) |
| 59 | \( 1 + 9.13e9iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 2.14e10iT - 2.65e21T^{2} \) |
| 67 | \( 1 - 6.42e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + 9.11e10T + 1.64e22T^{2} \) |
| 73 | \( 1 - 2.08e11iT - 2.29e22T^{2} \) |
| 79 | \( 1 + 3.05e11T + 5.90e22T^{2} \) |
| 83 | \( 1 - 2.37e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 - 1.12e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 9.61e11iT - 6.93e23T^{2} \) |
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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
−11.65270674788098028841958027595, −11.24883828306907822272433265651, −10.08704283004031787116948584887, −9.572406196198083535273854824177, −8.573788836240413847454546670480, −7.00131857013464772734494702349, −5.83085631316053455377201072594, −4.28301188490156823054516633172, −3.42778897862782658297362498294, −2.09947712378431270587797683499,
0.11685158603790761707955465875, 0.940844337673965739961858150925, 1.69266300192963976662731350222, 3.03767413397118104860404323411, 5.30446962038805502059606463749, 6.36748176456377032763999167482, 7.48633950016250756008866115487, 8.290786445900289629137481464276, 9.047248796400692316002471510193, 10.71256100710380010192482284610
