| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0477062 + 0.300587i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0477062 + 0.300587i\) |
| \(L(\frac{3}{2})\) |
|
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
−10.58146408842394759270215540037, −10.18820040222772632869534083371, −9.547200219348599132115564211142, −8.228961192136410297657798424676, −7.44153617178046510531622099232, −6.73728729498086499537741807549, −5.92446901953642313495905722234, −4.26972965883368255345978709413, −3.20522168714599649698801888938, −1.95025151170830758376888678804,
0.19368696061180433611090506274, 1.97519398733150736980715138839, 3.23798081802239431159607169592, 4.79638243058840551939310776857, 5.67451495033424106054574451189, 6.67689601304879097051879654202, 7.81821112938596264518928243751, 8.538780949948116020498419233732, 9.489765687736881830070967946987, 9.739535368163759787081607187671
