L(s) = 1 | + (1.56 − 2.70i)5-s + (−17.1 − 6.88i)7-s + (33.2 − 19.2i)11-s − 13.8i·13-s + (47.5 + 82.4i)17-s + (−9.86 − 5.69i)19-s + (23.9 + 13.8i)23-s + (57.6 + 99.7i)25-s − 44.2i·29-s + (119. − 68.7i)31-s + (−45.4 + 35.7i)35-s + (70.6 − 122. i)37-s − 337.·41-s − 417.·43-s + (145. − 251. i)47-s + ⋯ |
L(s) = 1 | + (0.139 − 0.242i)5-s + (−0.928 − 0.371i)7-s + (0.912 − 0.526i)11-s − 0.294i·13-s + (0.678 + 1.17i)17-s + (−0.119 − 0.0687i)19-s + (0.217 + 0.125i)23-s + (0.460 + 0.798i)25-s − 0.283i·29-s + (0.690 − 0.398i)31-s + (−0.219 + 0.172i)35-s + (0.313 − 0.543i)37-s − 1.28·41-s − 1.48·43-s + (0.450 − 0.779i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.673650457\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.673650457\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (17.1 + 6.88i)T \) |
good | 5 | \( 1 + (-1.56 + 2.70i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-33.2 + 19.2i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 13.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-47.5 - 82.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (9.86 + 5.69i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-23.9 - 13.8i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 44.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-119. + 68.7i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-70.6 + 122. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 417.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-145. + 251. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-14.7 + 8.53i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (299. + 519. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-459. - 265. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (325. + 563. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 934. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-787. + 454. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-397. + 688. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 314.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (179. - 310. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 80.5iT - 9.12e5T^{2} \) |
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
−9.417191021267221138931494324682, −8.638595267498229929468733074869, −7.76527424420248294763722967334, −6.66591570952730341521839325711, −6.12650386038082927614876720960, −5.08771194393254614151482353015, −3.81296473484918978066094444127, −3.23997291807927105035939234700, −1.64933024222852645521514707237, −0.48372839497376609065173246638,
1.06521378450953047207374890609, 2.50144091408982051211249129800, 3.36620565845820552226092074368, 4.50771388803979975708540349563, 5.50805330955466335002366490398, 6.70546442515159934626177103250, 6.84930499439699842961383894376, 8.210940771818472815003188419451, 9.089462001243653060560406258175, 9.779238577700329537077511139345