| L(s) = 1 | + (2.52 + 1.72i)3-s + (−0.607 − 0.0455i)5-s + (−2.56 + 0.663i)7-s + (2.32 + 5.91i)9-s + (5.84 + 2.29i)11-s + (−1.03 + 0.828i)13-s + (−1.45 − 1.16i)15-s + (−1.89 − 2.03i)17-s + (−2.73 + 4.73i)19-s + (−7.61 − 2.73i)21-s + (4.76 − 5.13i)23-s + (−4.57 − 0.689i)25-s + (−2.28 + 9.99i)27-s + (1.81 + 7.93i)29-s + (−2.65 − 4.59i)31-s + ⋯ |
| L(s) = 1 | + (1.45 + 0.994i)3-s + (−0.271 − 0.0203i)5-s + (−0.968 + 0.250i)7-s + (0.773 + 1.97i)9-s + (1.76 + 0.691i)11-s + (−0.288 + 0.229i)13-s + (−0.376 − 0.300i)15-s + (−0.459 − 0.494i)17-s + (−0.626 + 1.08i)19-s + (−1.66 − 0.596i)21-s + (0.994 − 1.07i)23-s + (−0.915 − 0.137i)25-s + (−0.439 + 1.92i)27-s + (0.336 + 1.47i)29-s + (−0.476 − 0.825i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0369 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0369 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54862 + 1.60695i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54862 + 1.60695i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.56 - 0.663i)T \) |
| good | 3 | \( 1 + (-2.52 - 1.72i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (0.607 + 0.0455i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-5.84 - 2.29i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (1.03 - 0.828i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (1.89 + 2.03i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (2.73 - 4.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.76 + 5.13i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.81 - 7.93i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (2.65 + 4.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.69 - 2.06i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-1.41 - 2.94i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.22 + 2.54i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-0.120 + 0.0181i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (5.13 - 1.58i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.675 + 9.01i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (0.474 - 1.53i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-8.76 + 5.06i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.66 + 1.29i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.427 + 2.83i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-8.97 - 5.18i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.22 + 10.3i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (8.82 - 3.46i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 15.8iT - 97T^{2} \) |
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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
−10.13527686605452203966664756462, −9.428771559521672701375768661952, −9.101395345320511924564402279644, −8.225566952090768940658011230467, −7.13522660716500684373292060145, −6.29599557834071066616214896766, −4.65181991262820197851616245318, −3.99707486291772171182291388543, −3.17804750577803871176765588567, −2.03320839009919156580597723722,
1.00899933377732589939309647095, 2.42057094029592171713823496093, 3.45054701876416441127924723023, 4.09807873549052090238951033146, 6.11647248020182000046164196431, 6.76347348830662791268781126655, 7.47041205759644076683871404365, 8.423555501050037300994164360696, 9.213046055890245740195310594020, 9.543381212584613486802289257914
