| L(s) = 1 | − 1.38i·3-s − 1.38i·5-s + 2.44·7-s + 1.07·9-s + 1.05i·11-s − i·13-s − 1.92·15-s + 6.96·17-s − 0.408i·19-s − 3.38i·21-s − 3.57·23-s + 3.07·25-s − 5.65i·27-s + 4.34i·29-s − 5.74·31-s + ⋯ |
| L(s) = 1 | − 0.800i·3-s − 0.620i·5-s + 0.923·7-s + 0.359·9-s + 0.318i·11-s − 0.277i·13-s − 0.496·15-s + 1.68·17-s − 0.0936i·19-s − 0.738i·21-s − 0.745·23-s + 0.615·25-s − 1.08i·27-s + 0.807i·29-s − 1.03·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45557 - 1.11689i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45557 - 1.11689i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.38iT - 3T^{2} \) |
| 5 | \( 1 + 1.38iT - 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 - 1.05iT - 11T^{2} \) |
| 17 | \( 1 - 6.96T + 17T^{2} \) |
| 19 | \( 1 + 0.408iT - 19T^{2} \) |
| 23 | \( 1 + 3.57T + 23T^{2} \) |
| 29 | \( 1 - 4.34iT - 29T^{2} \) |
| 31 | \( 1 + 5.74T + 31T^{2} \) |
| 37 | \( 1 + 7.49iT - 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 - 2.27iT - 43T^{2} \) |
| 47 | \( 1 + 4.33T + 47T^{2} \) |
| 53 | \( 1 - 0.647iT - 53T^{2} \) |
| 59 | \( 1 + 3.82iT - 59T^{2} \) |
| 61 | \( 1 + 4.80iT - 61T^{2} \) |
| 67 | \( 1 + 11.4iT - 67T^{2} \) |
| 71 | \( 1 - 7.66T + 71T^{2} \) |
| 73 | \( 1 - 7.68T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 15.4iT - 83T^{2} \) |
| 89 | \( 1 - 2.88T + 89T^{2} \) |
| 97 | \( 1 + 3.30T + 97T^{2} \) |
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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
−10.01979146695002273605383457734, −9.129951266487363162958040371034, −8.026208767634164433928393834995, −7.70394674809615877933823892464, −6.70727147541810496173715077695, −5.52169630952857431434701227925, −4.82249007142728506324253641278, −3.59557826881958686699501212406, −1.98077737821265871394525418317, −1.07264980885902754452969912840,
1.55174146393641127510672517252, 3.10036894029917550376563716231, 4.03231680741693088445531431536, 4.98737190882901598762158005572, 5.86611717443675682642305974798, 7.05894566303804426866535551331, 7.85692838227937723839263351282, 8.708588320409976245925598949631, 9.876722214269646191204184542179, 10.20950519715333182692145632490
