| L(s) = 1 | + (−0.654 + 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.415 + 0.909i)5-s + (−0.142 + 0.989i)6-s + (−2.83 + 0.832i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)10-s + (−1.08 − 1.24i)11-s + (−0.654 − 0.755i)12-s + (6.41 + 1.88i)13-s + (1.22 − 2.68i)14-s + (0.841 + 0.540i)15-s + (−0.959 + 0.281i)16-s + (−1.04 + 7.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.185 + 0.406i)5-s + (−0.0580 + 0.404i)6-s + (−1.07 + 0.314i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.303 − 0.0890i)10-s + (−0.325 − 0.375i)11-s + (−0.189 − 0.218i)12-s + (1.77 + 0.522i)13-s + (0.328 − 0.718i)14-s + (0.217 + 0.139i)15-s + (−0.239 + 0.0704i)16-s + (−0.252 + 1.75i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.981298 + 0.789861i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.981298 + 0.789861i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-4.77 + 0.393i)T \) |
| good | 7 | \( 1 + (2.83 - 0.832i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (1.08 + 1.24i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-6.41 - 1.88i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (1.04 - 7.24i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.0331 - 0.230i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (1.22 - 8.50i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.547 - 0.352i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-2.22 + 4.88i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-3.35 - 7.34i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.11 + 1.35i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 + (0.0131 - 0.00385i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-6.55 - 1.92i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (9.37 + 6.02i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.687 + 0.793i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-1.76 + 2.04i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.13 - 7.87i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (6.77 + 1.99i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (1.45 - 3.18i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (5.25 - 3.37i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (2.63 + 5.76i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.68412314774732726066996364418, −9.528157592685060018934010165502, −8.816063509299946253764069798241, −8.251460783745804528936486219602, −7.02745974889014260721911694619, −6.32787119900560923928936654827, −5.74569919034537848753193388526, −3.95162055715183793225178329651, −3.00455198466536659627507612436, −1.47220937540030407620119919161,
0.804406632487085956781206723371, 2.56808026574192793455010698324, 3.45558083098505104496207955741, 4.49715727282698708202851094006, 5.78479142138545993395067277909, 6.93065109822569945790782833690, 7.86822411542204353302228576452, 8.865834331489109893824970946781, 9.414957449798931796115567327170, 10.13567747039534598551335603536
