L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 4.44i·7-s − i·8-s − 9-s + 3.44·11-s + i·12-s + 2.44i·13-s − 4.44·14-s + 16-s + 4.44i·17-s − i·18-s − 19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.68i·7-s − 0.353i·8-s − 0.333·9-s + 1.04·11-s + 0.288i·12-s + 0.679i·13-s − 1.18·14-s + 0.250·16-s + 1.07i·17-s − 0.235i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.323721446\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.323721446\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 4.44iT - 7T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 4.44iT - 17T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 - 4.34T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 7.79iT - 37T^{2} \) |
| 41 | \( 1 + 0.898T + 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 + 7.79iT - 47T^{2} \) |
| 53 | \( 1 - 7.44iT - 53T^{2} \) |
| 59 | \( 1 + 6.44T + 59T^{2} \) |
| 61 | \( 1 + 9.44T + 61T^{2} \) |
| 67 | \( 1 - 15.2iT - 67T^{2} \) |
| 71 | \( 1 + 1.55T + 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + 8.34iT - 83T^{2} \) |
| 89 | \( 1 + 2.10T + 89T^{2} \) |
| 97 | \( 1 - 1.55iT - 97T^{2} \) |
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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
−8.926771991773861332871833703878, −8.462268874726520320946898207238, −7.56422800552519759352419560156, −6.68329434818883010966208033551, −6.10417869607014814189160515410, −5.59913474054024382985214248841, −4.54905551092051963536742248645, −3.58932012627572084690253009848, −2.37804710498169527703026368677, −1.48557425349802228076577527594,
0.44571615111613008040360151820, 1.44413451683206298126768884943, 2.96578955765466238624952974586, 3.60878199440830673893677074451, 4.44885513547987993140313032825, 4.92961659937106844121984856254, 6.20235042795367898748162237213, 6.98309983669833061321834187475, 7.79437792589739788342940528488, 8.587521400340495970213413195452