| L(s) = 1 | + (2 + 2i)2-s + 0.899·3-s + 8i·4-s + (23.4 + 23.4i)5-s + (1.79 + 1.79i)6-s + (14.3 − 14.3i)7-s + (−16 + 16i)8-s − 80.1·9-s + 93.6i·10-s + (100. − 100. i)11-s + 7.19i·12-s + (−75.6 − 151. i)13-s + 57.2·14-s + (21.0 + 21.0i)15-s − 64·16-s − 132. i·17-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.5i)2-s + 0.0999·3-s + 0.5i·4-s + (0.936 + 0.936i)5-s + (0.0499 + 0.0499i)6-s + (0.292 − 0.292i)7-s + (−0.250 + 0.250i)8-s − 0.990·9-s + 0.936i·10-s + (0.833 − 0.833i)11-s + 0.0499i·12-s + (−0.447 − 0.894i)13-s + 0.292·14-s + (0.0935 + 0.0935i)15-s − 0.250·16-s − 0.457i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.65768 + 0.858057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65768 + 0.858057i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 - 2i)T \) |
| 13 | \( 1 + (75.6 + 151. i)T \) |
| good | 3 | \( 1 - 0.899T + 81T^{2} \) |
| 5 | \( 1 + (-23.4 - 23.4i)T + 625iT^{2} \) |
| 7 | \( 1 + (-14.3 + 14.3i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 + (-100. + 100. i)T - 1.46e4iT^{2} \) |
| 17 | \( 1 + 132. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + (153. + 153. i)T + 1.30e5iT^{2} \) |
| 23 | \( 1 - 428. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 37.8T + 7.07e5T^{2} \) |
| 31 | \( 1 + (-1.25e3 - 1.25e3i)T + 9.23e5iT^{2} \) |
| 37 | \( 1 + (1.35e3 - 1.35e3i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 + (1.82e3 + 1.82e3i)T + 2.82e6iT^{2} \) |
| 43 | \( 1 + 1.40e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-1.70e3 + 1.70e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + 722.T + 7.89e6T^{2} \) |
| 59 | \( 1 + (2.10e3 - 2.10e3i)T - 1.21e7iT^{2} \) |
| 61 | \( 1 - 2.49e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-5.33e3 - 5.33e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + (6.40e3 + 6.40e3i)T + 2.54e7iT^{2} \) |
| 73 | \( 1 + (6.70e3 - 6.70e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 1.75e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (-2.13e3 - 2.13e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + (5.95e3 - 5.95e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (2.81e3 + 2.81e3i)T + 8.85e7iT^{2} \) |
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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
−17.13140678396601189237532619574, −15.37609211372429854008234825337, −14.16301772444498668977439687707, −13.72664616986314453299930426324, −11.78444026991222291525223315598, −10.41769604308446668871881349797, −8.635295926768755179814518516568, −6.83917883023644842288162573839, −5.53659633690668285959017047613, −3.03101956091323435108305190412,
1.95699641220802014260291680897, 4.62396427643908530683160710504, 6.14917448899928043985933350723, 8.717042809932116269922512860916, 9.817883987944773312900126492611, 11.63808025732807783424145792082, 12.64264079675286642827475958978, 13.97913062567809776037358247972, 14.84598763562741714028382202174, 16.77839857015640889400867458251
