| L(s) = 1 | − 3-s + 2.49i·5-s + 1.10i·7-s + 9-s − 3.60i·11-s + (3.49 + 0.890i)13-s − 2.49i·15-s + 2·17-s − 1.10i·19-s − 1.10i·21-s + 7.20·23-s − 1.21·25-s − 27-s − 5.20·29-s + 6.09i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.11i·5-s + 0.419i·7-s + 0.333·9-s − 1.08i·11-s + (0.969 + 0.246i)13-s − 0.643i·15-s + 0.485·17-s − 0.254i·19-s − 0.242i·21-s + 1.50·23-s − 0.243·25-s − 0.192·27-s − 0.967·29-s + 1.09i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.347256495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.347256495\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + (-3.49 - 0.890i)T \) |
| good | 5 | \( 1 - 2.49iT - 5T^{2} \) |
| 7 | \( 1 - 1.10iT - 7T^{2} \) |
| 11 | \( 1 + 3.60iT - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 1.10iT - 19T^{2} \) |
| 23 | \( 1 - 7.20T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 - 6.09iT - 31T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 - 7.48iT - 41T^{2} \) |
| 43 | \( 1 + 9.42T + 43T^{2} \) |
| 47 | \( 1 + 7.60iT - 47T^{2} \) |
| 53 | \( 1 - 4.76T + 53T^{2} \) |
| 59 | \( 1 - 1.38iT - 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 11.8iT - 67T^{2} \) |
| 71 | \( 1 - 1.82iT - 71T^{2} \) |
| 73 | \( 1 - 4.98iT - 73T^{2} \) |
| 79 | \( 1 + 0.439T + 79T^{2} \) |
| 83 | \( 1 + 0.591iT - 83T^{2} \) |
| 89 | \( 1 - 1.06iT - 89T^{2} \) |
| 97 | \( 1 + 3.01iT - 97T^{2} \) |
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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
−10.12240309108824859812475671952, −8.967347418976586733914923765467, −8.353079019464650468234607019464, −7.14283827001234574397157330551, −6.55873920638537307483292575308, −5.80442068653942007049061508873, −4.93813895728369550288436040342, −3.51206941367660993902140468312, −2.91960384103470356419451528762, −1.26320948420147371728799753464,
0.72122592559180818225806560712, 1.82690241257789662699511752811, 3.58988484958230929772264601872, 4.48663878899364743647034754127, 5.25727540235484917596310588588, 6.03055489127791168749034300292, 7.17480933641330542266114166944, 7.77065290420710599133767032720, 8.918375647141317763817269241475, 9.392748365313223650703275351984
