L(s) = 1 | + (10.1 + 17.5i)5-s + (−12.0 + 14.0i)7-s + (−15.4 − 8.92i)11-s + 33.1i·13-s + (22.9 − 39.6i)17-s + (35.0 − 20.2i)19-s + (−69.7 + 40.2i)23-s + (−142. + 246. i)25-s + 233. i·29-s + (195. + 112. i)31-s + (−368. − 68.3i)35-s + (135. + 234. i)37-s + 154.·41-s − 367.·43-s + (−263. − 457. i)47-s + ⋯ |
L(s) = 1 | + (0.905 + 1.56i)5-s + (−0.649 + 0.760i)7-s + (−0.423 − 0.244i)11-s + 0.707i·13-s + (0.326 − 0.566i)17-s + (0.422 − 0.244i)19-s + (−0.632 + 0.365i)23-s + (−1.14 + 1.97i)25-s + 1.49i·29-s + (1.13 + 0.653i)31-s + (−1.78 − 0.329i)35-s + (0.601 + 1.04i)37-s + 0.588·41-s − 1.30·43-s + (−0.818 − 1.41i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0354i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.400297374\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.400297374\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (12.0 - 14.0i)T \) |
good | 5 | \( 1 + (-10.1 - 17.5i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (15.4 + 8.92i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 33.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-22.9 + 39.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-35.0 + 20.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (69.7 - 40.2i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 233. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-195. - 112. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-135. - 234. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (263. + 457. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (78.8 + 45.5i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (312. - 541. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (78.8 - 45.5i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-431. + 747. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 303. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (999. + 576. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (3.48 + 6.02i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 815.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (155. + 269. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.83e3iT - 9.12e5T^{2} \) |
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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
−9.939696369675179282319214081694, −9.446776905925142394563223908268, −8.412373052351162715501856964006, −7.20033576276027054558725024443, −6.60603875532187557084438277446, −5.91062478653767023366292039371, −5.00021963456417224615535759117, −3.28514906631008915060324768068, −2.85022948481407598331787150786, −1.74431603761100430087874516399,
0.34300000304872457911850922919, 1.27177480811660621477425077068, 2.53650384261539502522205195776, 3.96943301916252746034159299412, 4.77186764487867519385670019666, 5.76905283466206594367628818431, 6.31978106439247155709353093341, 7.81702008333059663399281809195, 8.188420135222882204023198549710, 9.432799059346628294832429852703