L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.799 − 1.53i)3-s + 1.00i·4-s + (−1.65 + 0.521i)6-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−1.72 − 2.45i)9-s − 1.70i·11-s + (1.53 + 0.799i)12-s + (−0.921 − 0.921i)13-s − 1.00·14-s − 1.00·16-s + (−4.76 − 4.76i)17-s + (−0.519 + 2.95i)18-s + 5.94i·19-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.461 − 0.887i)3-s + 0.500i·4-s + (−0.674 + 0.212i)6-s + (0.267 − 0.267i)7-s + (0.250 − 0.250i)8-s + (−0.574 − 0.818i)9-s − 0.514i·11-s + (0.443 + 0.230i)12-s + (−0.255 − 0.255i)13-s − 0.267·14-s − 0.250·16-s + (−1.15 − 1.15i)17-s + (−0.122 + 0.696i)18-s + 1.36i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.033666241\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.033666241\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.799 + 1.53i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 + 1.70iT - 11T^{2} \) |
| 13 | \( 1 + (0.921 + 0.921i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.76 + 4.76i)T + 17iT^{2} \) |
| 19 | \( 1 - 5.94iT - 19T^{2} \) |
| 23 | \( 1 + (-2.49 + 2.49i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.19T + 29T^{2} \) |
| 31 | \( 1 + 3.40T + 31T^{2} \) |
| 37 | \( 1 + (-1.02 + 1.02i)T - 37iT^{2} \) |
| 41 | \( 1 + 10.9iT - 41T^{2} \) |
| 43 | \( 1 + (8.17 + 8.17i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.436 + 0.436i)T + 47iT^{2} \) |
| 53 | \( 1 + (6.87 - 6.87i)T - 53iT^{2} \) |
| 59 | \( 1 + 0.686T + 59T^{2} \) |
| 61 | \( 1 + 1.74T + 61T^{2} \) |
| 67 | \( 1 + (1.03 - 1.03i)T - 67iT^{2} \) |
| 71 | \( 1 + 12.2iT - 71T^{2} \) |
| 73 | \( 1 + (-4.59 - 4.59i)T + 73iT^{2} \) |
| 79 | \( 1 - 7.19iT - 79T^{2} \) |
| 83 | \( 1 + (-6.64 + 6.64i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.25T + 89T^{2} \) |
| 97 | \( 1 + (13.2 - 13.2i)T - 97iT^{2} \) |
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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
−9.314233889003379695629669044865, −8.700063347263782832704770516033, −7.947276847681132108792484644167, −7.18301080225675292142676893724, −6.40736399221463314890403943490, −5.17007030993422279740581657421, −3.85043696192889743745293618591, −2.83778984834012669024273281149, −1.84651402869119857066318345976, −0.49985075431547244181652761874,
1.87332996067805828101616396857, 3.04844002993609285468643967725, 4.50665691676578615088671761119, 4.90460334939743795244324710999, 6.18554509064621366325762649543, 7.02808744050373979783095014138, 8.108718992621019050703096864589, 8.667040019000020928921822895800, 9.450137886477163636355448498264, 10.01794697220761521024643109563