L(s) = 1 | + (6.38 + 11.0i)5-s + (−2.53 − 18.3i)7-s + (46.8 + 27.0i)11-s + 8.85i·13-s + (−34.4 + 59.6i)17-s + (141. − 81.9i)19-s + (−81.3 + 46.9i)23-s + (−18.9 + 32.8i)25-s + 119. i·29-s + (−85.6 − 49.4i)31-s + (186. − 145. i)35-s + (−47.0 − 81.5i)37-s + 259.·41-s − 5.01·43-s + (−28.6 − 49.6i)47-s + ⋯ |
L(s) = 1 | + (0.570 + 0.988i)5-s + (−0.137 − 0.990i)7-s + (1.28 + 0.741i)11-s + 0.188i·13-s + (−0.491 + 0.851i)17-s + (1.71 − 0.989i)19-s + (−0.737 + 0.425i)23-s + (−0.151 + 0.262i)25-s + 0.765i·29-s + (−0.496 − 0.286i)31-s + (0.901 − 0.700i)35-s + (−0.209 − 0.362i)37-s + 0.987·41-s − 0.0177·43-s + (−0.0889 − 0.154i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.525676763\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.525676763\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (2.53 + 18.3i)T \) |
good | 5 | \( 1 + (-6.38 - 11.0i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-46.8 - 27.0i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 8.85iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (34.4 - 59.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-141. + 81.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (81.3 - 46.9i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 119. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (85.6 + 49.4i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (47.0 + 81.5i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 259.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 5.01T + 7.95e4T^{2} \) |
| 47 | \( 1 + (28.6 + 49.6i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-407. - 235. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-112. + 195. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-370. + 213. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-81.9 + 141. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 79.8iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-666. - 384. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-267. - 463. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 438.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-12.8 - 22.2i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.38e3iT - 9.12e5T^{2} \) |
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
−9.656439819878640892310523975720, −9.203797855849677409990674930771, −7.81339824163464863673302758507, −6.91180168675228060935348614734, −6.65626691106585597751184782079, −5.47163045428666195810168130888, −4.21202557833419635801408804251, −3.49092431163315519302533645381, −2.21306730694804579159854159416, −1.08155147237021322342472759024,
0.74192405137990296685819672562, 1.77989376435194340466176270890, 3.04881101787818497175047768596, 4.18271934248755108334327831506, 5.40399860979940814906275966528, 5.77528916991061827666544988002, 6.80225735462509141496770741374, 8.024461383624443351538014414706, 8.816576606144786800187948832249, 9.370028002739207280656140272249