| L(s) = 1 | + (0.345 + 1.37i)2-s + (−0.245 − 1.71i)3-s + (−1.76 + 0.948i)4-s + (0.286 − 0.326i)5-s + (2.26 − 0.929i)6-s + (−0.859 + 0.113i)7-s + (−1.90 − 2.08i)8-s + (−2.87 + 0.843i)9-s + (0.547 + 0.280i)10-s + (−2.07 + 4.20i)11-s + (2.05 + 2.78i)12-s + (−1.47 + 4.33i)13-s + (−0.452 − 1.13i)14-s + (−0.631 − 0.411i)15-s + (2.20 − 3.33i)16-s + (2.54 − 2.54i)17-s + ⋯ |
| L(s) = 1 | + (0.244 + 0.969i)2-s + (−0.141 − 0.989i)3-s + (−0.880 + 0.474i)4-s + (0.128 − 0.146i)5-s + (0.925 − 0.379i)6-s + (−0.324 + 0.0427i)7-s + (−0.674 − 0.737i)8-s + (−0.959 + 0.281i)9-s + (0.173 + 0.0885i)10-s + (−0.624 + 1.26i)11-s + (0.594 + 0.804i)12-s + (−0.408 + 1.20i)13-s + (−0.120 − 0.304i)14-s + (−0.162 − 0.106i)15-s + (0.550 − 0.834i)16-s + (0.616 − 0.616i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.220000 + 0.738872i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.220000 + 0.738872i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.345 - 1.37i)T \) |
| 3 | \( 1 + (0.245 + 1.71i)T \) |
| good | 5 | \( 1 + (-0.286 + 0.326i)T + (-0.652 - 4.95i)T^{2} \) |
| 7 | \( 1 + (0.859 - 0.113i)T + (6.76 - 1.81i)T^{2} \) |
| 11 | \( 1 + (2.07 - 4.20i)T + (-6.69 - 8.72i)T^{2} \) |
| 13 | \( 1 + (1.47 - 4.33i)T + (-10.3 - 7.91i)T^{2} \) |
| 17 | \( 1 + (-2.54 + 2.54i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.422 - 2.12i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (1.19 - 9.03i)T + (-22.2 - 5.95i)T^{2} \) |
| 29 | \( 1 + (-3.70 - 0.242i)T + (28.7 + 3.78i)T^{2} \) |
| 31 | \( 1 + (7.63 + 4.40i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.03 + 5.19i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (0.367 - 2.78i)T + (-39.6 - 10.6i)T^{2} \) |
| 43 | \( 1 + (3.66 + 1.80i)T + (26.1 + 34.1i)T^{2} \) |
| 47 | \( 1 + (11.0 + 2.97i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.13 - 4.68i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-9.74 + 11.1i)T + (-7.70 - 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.568 + 8.66i)T + (-60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (4.61 - 2.27i)T + (40.7 - 53.1i)T^{2} \) |
| 71 | \( 1 + (2.19 - 5.30i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (2.08 + 5.04i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (3.27 - 12.2i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-7.58 + 6.65i)T + (10.8 - 82.2i)T^{2} \) |
| 89 | \( 1 + (-11.7 - 4.87i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (10.7 - 6.19i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.42706133734172838606326513649, −9.721322447281456748261325413260, −9.362869160970418705462760140561, −8.012396596996489261597123441989, −7.35849212454819441715043371832, −6.79443220262484458442975343872, −5.63058046506098169438431836630, −4.95234484656160555738724072460, −3.48412882804820827773557822228, −1.86677539502678981387788186587,
0.39925493900069952089762316922, 2.77393287410979538465882575815, 3.33712637829426569635977078180, 4.61973933276952718783606718862, 5.47647586969551554789890369410, 6.29319528369746171891593649424, 8.263090231297850023279802511759, 8.671868099086324814323496542262, 10.02334407904806476411849266606, 10.34197087848895444130672834087
