L(s) = 1 | + (−4.07 − 3.21i)3-s + (0.330 − 0.571i)5-s + (−15.6 − 9.91i)7-s + (6.26 + 26.2i)9-s + (21.4 − 12.3i)11-s + (−43.5 − 25.1i)13-s + (−3.18 + 1.26i)15-s − 67.5·17-s + 62.9i·19-s + (31.8 + 90.7i)21-s + (135. + 78.4i)23-s + (62.2 + 107. i)25-s + (58.9 − 127. i)27-s + (−129. + 74.9i)29-s + (139. + 80.5i)31-s + ⋯ |
L(s) = 1 | + (−0.784 − 0.619i)3-s + (0.0295 − 0.0511i)5-s + (−0.844 − 0.535i)7-s + (0.232 + 0.972i)9-s + (0.586 − 0.338i)11-s + (−0.928 − 0.536i)13-s + (−0.0548 + 0.0218i)15-s − 0.963·17-s + 0.760i·19-s + (0.331 + 0.943i)21-s + (1.23 + 0.710i)23-s + (0.498 + 0.863i)25-s + (0.420 − 0.907i)27-s + (−0.831 + 0.480i)29-s + (0.808 + 0.466i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5351783491\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5351783491\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.07 + 3.21i)T \) |
| 7 | \( 1 + (15.6 + 9.91i)T \) |
good | 5 | \( 1 + (-0.330 + 0.571i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-21.4 + 12.3i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (43.5 + 25.1i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 67.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 62.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-135. - 78.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (129. - 74.9i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-139. - 80.5i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 16.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + (134. - 233. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-188. - 325. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (31.3 + 54.3i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 136. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (358. - 621. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-23.9 + 13.8i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (163. - 283. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 246. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 261. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (391. + 678. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (599. + 1.03e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 968.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.10e3 + 639. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.81179123060469507912979987251, −10.96025734332655462986984235370, −10.04685904293755299251799528034, −8.982396293126953165104562943148, −7.54811628997177338268448050269, −6.84342488420454024116087935857, −5.85272171435387414199254607629, −4.66958280250006484430199742211, −3.09178850888428330414768438590, −1.24287234271101923221212472028,
0.25549331700906885804119166899, 2.55449782708942853835554599734, 4.13855306231793643215086422129, 5.09283265271576639107167509518, 6.42016731426787973437565568052, 6.97305491309792781892145973941, 8.909215729971475781274213521259, 9.439361467181955871859357617431, 10.43173815157155151623513191489, 11.41291906943811153811328611421