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 | \( 1 + 1.38e3iT - 9.12e5T^{2} \) |
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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
−9.370028002739207280656140272249, −8.816576606144786800187948832249, −8.024461383624443351538014414706, −6.80225735462509141496770741374, −5.77528916991061827666544988002, −5.40399860979940814906275966528, −4.18271934248755108334327831506, −3.04881101787818497175047768596, −1.77989376435194340466176270890, −0.74192405137990296685819672562,
1.08155147237021322342472759024, 2.21306730694804579159854159416, 3.49092431163315519302533645381, 4.21202557833419635801408804251, 5.47163045428666195810168130888, 6.65626691106585597751184782079, 6.91180168675228060935348614734, 7.81339824163464863673302758507, 9.203797855849677409990674930771, 9.656439819878640892310523975720