L(s) = 1 | + 2-s + 4-s − 2.23i·5-s + (−1.58 − 2.12i)7-s + 8-s − 2.23i·10-s − 1.41i·11-s − 3.16·13-s + (−1.58 − 2.12i)14-s + 16-s − 4.47i·17-s − 2.23i·20-s − 1.41i·22-s + 6·23-s − 5.00·25-s − 3.16·26-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.999i·5-s + (−0.597 − 0.801i)7-s + 0.353·8-s − 0.707i·10-s − 0.426i·11-s − 0.877·13-s + (−0.422 − 0.566i)14-s + 0.250·16-s − 1.08i·17-s − 0.499i·20-s − 0.301i·22-s + 1.25·23-s − 1.00·25-s − 0.620·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0250 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0250 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.38444 - 1.35020i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38444 - 1.35020i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (1.58 + 2.12i)T \) |
good | 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 4.47iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 6.32T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−10.41011251521418419867995766103, −9.543037458507303540981638483703, −8.715198393328260542566724410106, −7.47434382277426184010969386820, −6.87043759280679137682174790671, −5.57272756508628950437021374759, −4.84037393461193360756176021155, −3.86599417797632285484527579714, −2.67686656673647784230340749782, −0.844671356794809025110105068571,
2.22805605885252604452251272302, 3.04367145682432147600519099007, 4.19343185181301668597937393421, 5.43656664268181940483207931602, 6.28442495715401823956566949450, 7.03909844704061318053381462248, 7.957821889134095819650809281944, 9.270487077738847780034135588011, 10.02606774710143072380019158207, 10.93118300640526380279364321653