| L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (−0.841 − 0.540i)6-s + (−0.838 + 1.83i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (−5.73 − 1.68i)11-s + (0.959 + 0.281i)12-s + (−0.0306 − 0.0670i)13-s + (0.287 − 1.99i)14-s + (0.654 − 0.755i)15-s + (0.415 − 0.909i)16-s + (−0.312 − 0.200i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.0636 − 0.442i)5-s + (−0.343 − 0.220i)6-s + (−0.316 + 0.693i)7-s + (−0.231 + 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.131 + 0.287i)10-s + (−1.72 − 0.507i)11-s + (0.276 + 0.0813i)12-s + (−0.00849 − 0.0186i)13-s + (0.0767 − 0.533i)14-s + (0.169 − 0.195i)15-s + (0.103 − 0.227i)16-s + (−0.0757 − 0.0487i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0170568 - 0.177568i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0170568 - 0.177568i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 5 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-0.340 + 4.78i)T \) |
| good | 7 | \( 1 + (0.838 - 1.83i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (5.73 + 1.68i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (0.0306 + 0.0670i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (0.312 + 0.200i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (6.48 - 4.17i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (0.668 + 0.429i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (2.98 - 3.45i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.0113 - 0.0786i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.61 - 11.2i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (1.84 + 2.12i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 8.45T + 47T^{2} \) |
| 53 | \( 1 + (-2.07 + 4.54i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (4.25 + 9.30i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-3.03 + 3.50i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-5.57 + 1.63i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (6.26 - 1.84i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (10.0 - 6.43i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.92 + 4.22i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.38 + 9.61i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (9.26 + 10.6i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-2.04 - 14.1i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.57751959730562080723259379215, −10.09425290548774839455258891407, −9.057415071277051636179967761278, −8.338103056879181132062744956301, −7.912656281015269010104126733428, −6.48813628458125800592152293073, −5.59295931942204309430706750492, −4.65262222920469050157323612817, −3.14854174973327900397686194247, −2.12143937470916601533497861528,
0.10072976542345400261626545105, 2.04795814631186862031085716091, 2.96201019784478957642801540507, 4.22171266192398341094456010362, 5.65569364209816580562471382924, 6.92992150886590957428979889407, 7.38070672835385666598631196503, 8.211410933574723802814299528303, 9.146946951958852403227791306384, 10.14826965981010997063061703752
