| L(s) = 1 | + (−2.13 + 0.571i)2-s + (0.756 − 0.436i)4-s + (−2.78 − 4.15i)5-s + (2.73 − 6.44i)7-s + (4.88 − 4.88i)8-s + (8.31 + 7.26i)10-s + (4.69 + 8.12i)11-s + (−0.405 + 0.405i)13-s + (−2.15 + 15.3i)14-s + (−9.36 + 16.2i)16-s + (−2.82 + 10.5i)17-s + (−23.8 − 13.7i)19-s + (−3.91 − 1.92i)20-s + (−14.6 − 14.6i)22-s + (−1.02 − 3.81i)23-s + ⋯ |
| L(s) = 1 | + (−1.06 + 0.285i)2-s + (0.189 − 0.109i)4-s + (−0.557 − 0.830i)5-s + (0.390 − 0.920i)7-s + (0.610 − 0.610i)8-s + (0.831 + 0.726i)10-s + (0.426 + 0.738i)11-s + (−0.0311 + 0.0311i)13-s + (−0.153 + 1.09i)14-s + (−0.585 + 1.01i)16-s + (−0.166 + 0.621i)17-s + (−1.25 − 0.724i)19-s + (−0.195 − 0.0961i)20-s + (−0.665 − 0.665i)22-s + (−0.0445 − 0.166i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0292349 - 0.205519i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0292349 - 0.205519i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.78 + 4.15i)T \) |
| 7 | \( 1 + (-2.73 + 6.44i)T \) |
| good | 2 | \( 1 + (2.13 - 0.571i)T + (3.46 - 2i)T^{2} \) |
| 11 | \( 1 + (-4.69 - 8.12i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (0.405 - 0.405i)T - 169iT^{2} \) |
| 17 | \( 1 + (2.82 - 10.5i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (23.8 + 13.7i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (1.02 + 3.81i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 34.6iT - 841T^{2} \) |
| 31 | \( 1 + (11.8 + 20.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-25.6 + 6.86i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 54.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (47.3 - 47.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-16.9 + 4.55i)T + (1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (14.7 + 3.95i)T + (2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (90.3 - 52.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (12.5 - 21.7i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-0.364 + 1.36i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 86.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (84.9 + 22.7i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-128. - 74.1i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (35.5 - 35.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (130. + 75.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-74.8 - 74.8i)T + 9.40e3iT^{2} \) |
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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
−10.80031503805841733882433710246, −9.902527949747583959579053627802, −9.006026183405107196915053850585, −8.181759802240297879520878969558, −7.50307527484114768511266403922, −6.48712768901053103819015210980, −4.58479226045290216437356669520, −4.09125255193642154027505073229, −1.56261914601299204556590819134, −0.14663422873134912186902390392,
1.83542913224971620027655378764, 3.24960540124946060932887548907, 4.82682111279549667137989380612, 6.15488638798824262025722247103, 7.33353090901434580570854930926, 8.421035197076618496760884487921, 8.852722725665482561986468268493, 10.05877146452761972247674344947, 10.88404195865885482595706152288, 11.51267450330760586437865799006
