L(s) = 1 | + (−1.35 − 2.35i)5-s + (8.74 + 16.3i)7-s + (−8.39 − 4.84i)11-s + 67.7i·13-s + (−50.1 + 86.7i)17-s + (59.6 − 34.4i)19-s + (−126. + 73.3i)23-s + (58.8 − 101. i)25-s − 284. i·29-s + (−197. − 113. i)31-s + (26.5 − 42.7i)35-s + (150. + 261. i)37-s + 232.·41-s − 173.·43-s + (−191. − 331. i)47-s + ⋯ |
L(s) = 1 | + (−0.121 − 0.210i)5-s + (0.471 + 0.881i)7-s + (−0.229 − 0.132i)11-s + 1.44i·13-s + (−0.714 + 1.23i)17-s + (0.720 − 0.416i)19-s + (−1.15 + 0.664i)23-s + (0.470 − 0.814i)25-s − 1.82i·29-s + (−1.14 − 0.659i)31-s + (0.128 − 0.206i)35-s + (0.669 + 1.16i)37-s + 0.885·41-s − 0.613·43-s + (−0.594 − 1.02i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00279i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5006461787\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5006461787\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-8.74 - 16.3i)T \) |
good | 5 | \( 1 + (1.35 + 2.35i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (8.39 + 4.84i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 67.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (50.1 - 86.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-59.6 + 34.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (126. - 73.3i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 284. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (197. + 113. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-150. - 261. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 232.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 173.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (191. + 331. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-22.2 - 12.8i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (371. - 644. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (343. - 198. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-293. + 508. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 501. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-616. - 356. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (466. + 807. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 837.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (442. + 766. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.17e3iT - 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.825777250219455767655439648803, −9.145455089922006554211909240809, −8.354629105401294620154756601549, −7.67915106006701598539929010508, −6.43314909137710560556647103971, −5.83185727388075711845925755886, −4.66891256297917177613894643386, −3.96202078612277890570192160487, −2.46163831404266657632915133115, −1.63807431518090193020289664610,
0.12250449769442346571440497078, 1.33230947263639794842770087623, 2.79390425097919594640164415986, 3.70282402920594846504298826286, 4.86662928217174892806346258324, 5.53712181897861057063299089000, 6.85735837599134645677479899565, 7.48781445028996261245527063676, 8.160018679459703331146972449479, 9.234799335021155249129974257559