L(s) = 1 | + (0.973 − 5.10i)3-s + (−6.29 + 10.8i)5-s + (18.2 − 2.87i)7-s + (−25.1 − 9.94i)9-s + (−59.5 + 34.3i)11-s + (34.1 − 19.7i)13-s + (49.4 + 42.7i)15-s + (−38.3 + 66.4i)17-s + (−69.3 + 40.0i)19-s + (3.13 − 96.1i)21-s + (−134. − 77.7i)23-s + (−16.6 − 28.8i)25-s + (−75.1 + 118. i)27-s + (197. + 114. i)29-s + 145. i·31-s + ⋯ |
L(s) = 1 | + (0.187 − 0.982i)3-s + (−0.562 + 0.974i)5-s + (0.987 − 0.155i)7-s + (−0.929 − 0.368i)9-s + (−1.63 + 0.942i)11-s + (0.728 − 0.420i)13-s + (0.851 + 0.735i)15-s + (−0.547 + 0.948i)17-s + (−0.836 + 0.483i)19-s + (0.0325 − 0.999i)21-s + (−1.22 − 0.704i)23-s + (−0.133 − 0.230i)25-s + (−0.535 + 0.844i)27-s + (1.26 + 0.731i)29-s + 0.842i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8102206781\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8102206781\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.973 + 5.10i)T \) |
| 7 | \( 1 + (-18.2 + 2.87i)T \) |
good | 5 | \( 1 + (6.29 - 10.8i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (59.5 - 34.3i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-34.1 + 19.7i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (38.3 - 66.4i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (69.3 - 40.0i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (134. + 77.7i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-197. - 114. i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 145. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-74.1 - 128. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-27.9 - 48.3i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (109. - 188. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 109.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (257. + 148. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + 179.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 318. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 653.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 157. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (1.05e3 + 609. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 - 818.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-404. + 700. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-20.9 - 36.3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (161. + 93.1i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.97828919989741399561829162060, −10.72429317702264163999668674492, −10.50255452972634778348758258414, −8.365729011367091166222394099827, −8.038150304364127972251554285305, −7.05493513382860803599462873731, −6.03744698621982943953378708862, −4.54044887330585897449723816633, −2.96307690765775680791972928259, −1.78364785245363876452565648161,
0.29829881044114212446747180079, 2.50435388563123359396558559549, 4.13216475028248775959335289437, 4.88155407624439200031594172995, 5.84086064110298615904286029233, 7.919237144688902353060670348414, 8.367962929968332189577490552027, 9.206785615035961538467365255681, 10.53627162605526401827240174410, 11.22860202676477771647976653202