L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.860 − 1.50i)3-s + 1.00i·4-s + (−2.19 − 0.424i)5-s + (0.454 − 1.67i)6-s + (−0.358 + 0.358i)7-s + (−0.707 + 0.707i)8-s + (−1.52 + 2.58i)9-s + (−1.25 − 1.85i)10-s − 1.95i·11-s + (1.50 − 0.860i)12-s + (1.74 + 1.74i)13-s − 0.507·14-s + (1.24 + 3.66i)15-s − 1.00·16-s + (3.65 + 3.65i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.496 − 0.867i)3-s + 0.500i·4-s + (−0.981 − 0.189i)5-s + (0.185 − 0.682i)6-s + (−0.135 + 0.135i)7-s + (−0.250 + 0.250i)8-s + (−0.506 + 0.862i)9-s + (−0.395 − 0.585i)10-s − 0.589i·11-s + (0.433 − 0.248i)12-s + (0.482 + 0.482i)13-s − 0.135·14-s + (0.322 + 0.946i)15-s − 0.250·16-s + (0.887 + 0.887i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33063 + 0.382954i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33063 + 0.382954i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.860 + 1.50i)T \) |
| 5 | \( 1 + (2.19 + 0.424i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + (0.358 - 0.358i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.95iT - 11T^{2} \) |
| 13 | \( 1 + (-1.74 - 1.74i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.65 - 3.65i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.69iT - 19T^{2} \) |
| 23 | \( 1 + (-6.27 + 6.27i)T - 23iT^{2} \) |
| 29 | \( 1 - 8.51T + 29T^{2} \) |
| 37 | \( 1 + (1.92 - 1.92i)T - 37iT^{2} \) |
| 41 | \( 1 + 9.77iT - 41T^{2} \) |
| 43 | \( 1 + (1.51 + 1.51i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.14 - 4.14i)T + 47iT^{2} \) |
| 53 | \( 1 + (9.34 - 9.34i)T - 53iT^{2} \) |
| 59 | \( 1 - 1.87T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + (7.86 - 7.86i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.88iT - 71T^{2} \) |
| 73 | \( 1 + (-9.70 - 9.70i)T + 73iT^{2} \) |
| 79 | \( 1 - 1.35iT - 79T^{2} \) |
| 83 | \( 1 + (-2.46 + 2.46i)T - 83iT^{2} \) |
| 89 | \( 1 + 0.192T + 89T^{2} \) |
| 97 | \( 1 + (-1.01 + 1.01i)T - 97iT^{2} \) |
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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
−10.49778760417672479395359580528, −8.780542064500239789510703420709, −8.334707141322300818910051429607, −7.51399747629372722976583583008, −6.65027224637449628125399862729, −5.97016319856792015265837059298, −4.99361814466512290228582251592, −3.94992908350975113426938165180, −2.86510131504584895047020820916, −1.06646080234896642074550162243,
0.78442131759677839859011033103, 3.02458051125106845218852831966, 3.54921789154285824719688656069, 4.75158762399638577385767361641, 5.17167135418606453193576216650, 6.49154378796368830011115126148, 7.30445063111038597465722687751, 8.458633737086569719499809783186, 9.472020264418399555338028496982, 10.11332266659837215591186038167