L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (−1.57 + 1.59i)5-s − 1.00·6-s + (−1.37 + 1.37i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (−2.23 + 0.0144i)10-s + 0.0288i·11-s + (−0.707 − 0.707i)12-s + (0.711 − 0.711i)13-s − 1.95·14-s + (−0.0144 − 2.23i)15-s − 1.00·16-s + (−0.438 + 0.438i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (−0.702 + 0.711i)5-s − 0.408·6-s + (−0.521 + 0.521i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (−0.707 + 0.00456i)10-s + 0.00870i·11-s + (−0.204 − 0.204i)12-s + (0.197 − 0.197i)13-s − 0.521·14-s + (−0.00372 − 0.577i)15-s − 0.250·16-s + (−0.106 + 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 + 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.795 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.196018 - 0.580194i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.196018 - 0.580194i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.57 - 1.59i)T \) |
| 23 | \( 1 + (-3.06 + 3.68i)T \) |
good | 7 | \( 1 + (1.37 - 1.37i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.0288iT - 11T^{2} \) |
| 13 | \( 1 + (-0.711 + 0.711i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.438 - 0.438i)T - 17iT^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 29 | \( 1 - 3.97iT - 29T^{2} \) |
| 31 | \( 1 + 5.20T + 31T^{2} \) |
| 37 | \( 1 + (-1.46 + 1.46i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.17T + 41T^{2} \) |
| 43 | \( 1 + (5.83 + 5.83i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.90 - 6.90i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.73 - 2.73i)T + 53iT^{2} \) |
| 59 | \( 1 - 5.73iT - 59T^{2} \) |
| 61 | \( 1 - 5.09iT - 61T^{2} \) |
| 67 | \( 1 + (4.63 - 4.63i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.47T + 71T^{2} \) |
| 73 | \( 1 + (0.908 - 0.908i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.20T + 79T^{2} \) |
| 83 | \( 1 + (-9.95 - 9.95i)T + 83iT^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + (0.850 - 0.850i)T - 97iT^{2} \) |
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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
−10.84814012198733089441954561347, −10.46984994576004375048439389560, −9.070490973494946829201369200132, −8.414953907609109630935472350476, −7.21141079809310548792046956579, −6.51902896829597751430801386015, −5.72089827765318595244724918858, −4.54889617690822862868916947891, −3.68129885647758174558305961232, −2.62473534582787048417215670852,
0.27830410178625944645891126920, 1.78843829609144164227239981308, 3.43204757513484670400890535620, 4.30879296163198325907252009427, 5.23222849045662103817072782206, 6.35191320089072630997161146538, 7.15747050139769268549187519277, 8.232178927507864274361163952137, 9.124191379166156472179292576531, 10.16496715395311943259573920400