| L(s) = 1 | − 43.1i·2-s − 202. i·3-s − 1.34e3·4-s + 1.76e3·5-s − 8.73e3·6-s − 2.93e3·7-s + 3.61e4i·8-s − 2.13e4·9-s − 7.63e4i·10-s + 6.36e4i·11-s + 2.73e5i·12-s − 1.30e5·13-s + 1.26e5i·14-s − 3.58e5i·15-s + 8.67e5·16-s − 5.07e5i·17-s + ⋯ |
| L(s) = 1 | − 1.90i·2-s − 1.44i·3-s − 2.63·4-s + 1.26·5-s − 2.75·6-s − 0.462·7-s + 3.11i·8-s − 1.08·9-s − 2.41i·10-s + 1.31i·11-s + 3.80i·12-s − 1.26·13-s + 0.882i·14-s − 1.82i·15-s + 3.30·16-s − 1.47i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.841211 + 0.372967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.841211 + 0.372967i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (2.55e6 - 2.82e6i)T \) |
| good | 2 | \( 1 + 43.1iT - 512T^{2} \) |
| 3 | \( 1 + 202. iT - 1.96e4T^{2} \) |
| 5 | \( 1 - 1.76e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.93e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.36e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + 1.30e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.07e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 + 6.39e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + 1.08e6T + 1.80e12T^{2} \) |
| 31 | \( 1 + 2.11e5iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 1.92e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 4.40e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 3.63e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 1.95e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 - 1.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.59e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.63e7iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 1.56e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.05e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.38e6iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 3.55e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 - 4.09e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.88e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + 4.24e8iT - 7.60e17T^{2} \) |
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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
−13.37739711551797665488607486739, −12.63907070269147994924333845113, −11.79262477286627203581647589894, −10.03065429366733659819946720552, −9.303278103848481709779862877315, −7.14930119804666171400091220938, −5.06114685693110022358795136252, −2.52514040765980329916623482440, −1.89966139007992657772026729458, −0.36033284695318456238367966102,
3.90720848013337617289226986570, 5.46116445950476128764127224448, 6.17058968009872853368802060160, 8.248081223159733376937280267017, 9.546004530482132271833638208836, 10.14602376658659067513979768094, 13.09890239900084175744772805438, 14.29289854736589158876895689349, 14.96237844334504391062922796335, 16.30148144337213438846746909515
