L(s) = 1 | + (−3.41 + 5.91i)5-s + (−50.5 + 29.1i)11-s − 38.5i·13-s + (−16.1 − 27.9i)17-s + (−107. − 62.2i)19-s + (−174. − 100. i)23-s + (39.2 + 67.9i)25-s + 104. i·29-s + (240. − 138. i)31-s + (23.8 − 41.2i)37-s + 387.·41-s + 272.·43-s + (81.5 − 141. i)47-s + (−313. + 181. i)53-s − 398. i·55-s + ⋯ |
L(s) = 1 | + (−0.305 + 0.528i)5-s + (−1.38 + 0.799i)11-s − 0.822i·13-s + (−0.230 − 0.398i)17-s + (−1.30 − 0.751i)19-s + (−1.57 − 0.911i)23-s + (0.313 + 0.543i)25-s + 0.668i·29-s + (1.39 − 0.805i)31-s + (0.105 − 0.183i)37-s + 1.47·41-s + 0.966·43-s + (0.253 − 0.438i)47-s + (−0.813 + 0.469i)53-s − 0.976i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.157273923\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.157273923\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + (3.41 - 5.91i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (50.5 - 29.1i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 38.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (16.1 + 27.9i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (107. + 62.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (174. + 100. i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 104. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-240. + 138. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-23.8 + 41.2i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 272.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-81.5 + 141. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (313. - 181. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-105. - 183. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-202. - 117. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (262. + 454. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 348. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (465. - 268. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (362. - 628. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 392.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (430. - 744. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 978. iT - 9.12e5T^{2} \) |
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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
−8.950430765781947956289951545116, −8.006711376788452494437814339938, −7.57134226375180701430166423997, −6.64826589109098694589306355391, −5.82166186616752819680583304996, −4.80903082389288659471162524296, −4.11603475332265429479712464192, −2.74883015365971328918089051198, −2.32602409398479190411524967591, −0.53719540107026925174478833319,
0.44104839799206860300686691549, 1.81883608636843528159775702868, 2.80239958936391754475198939510, 4.06084151004276446497652089780, 4.58611141444196359345918629027, 5.81092514030520082228904596703, 6.23785049655381680701459342085, 7.50069202246513922441863964119, 8.259329839895689707571650077130, 8.570046639471749919761244005627