L(s) = 1 | + (−4.08 + 8.01i)3-s − 47.0i·5-s − 13.4·7-s + (−47.6 − 65.5i)9-s − 135. i·11-s − 80.8·13-s + (377. + 192. i)15-s − 430. i·17-s + 43.5·19-s + (54.9 − 107. i)21-s + 873. i·23-s − 1.59e3·25-s + (719. − 114. i)27-s + 707. i·29-s − 677.·31-s + ⋯ |
L(s) = 1 | + (−0.453 + 0.891i)3-s − 1.88i·5-s − 0.274·7-s + (−0.587 − 0.808i)9-s − 1.11i·11-s − 0.478·13-s + (1.67 + 0.854i)15-s − 1.49i·17-s + 0.120·19-s + (0.124 − 0.244i)21-s + 1.65i·23-s − 2.54·25-s + (0.987 − 0.156i)27-s + 0.840i·29-s − 0.705·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2331993176\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2331993176\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.08 - 8.01i)T \) |
good | 5 | \( 1 + 47.0iT - 625T^{2} \) |
| 7 | \( 1 + 13.4T + 2.40e3T^{2} \) |
| 11 | \( 1 + 135. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 80.8T + 2.85e4T^{2} \) |
| 17 | \( 1 + 430. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 43.5T + 1.30e5T^{2} \) |
| 23 | \( 1 - 873. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 707. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 677.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.63e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.22e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.43e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.06e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.46e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 470. iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 39.9T + 1.38e7T^{2} \) |
| 67 | \( 1 + 4.52e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 9.85e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.27e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.93e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.27e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 9.78e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.32e3T + 8.85e7T^{2} \) |
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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
−9.911664204758993927973617881564, −9.272521404236900368980374826340, −8.709413922896784471929007911370, −7.49916802350796771331324778118, −5.83462236259419267267155518472, −5.25704385871869841601715172784, −4.40171564591011950042915570656, −3.20226223734340609099364553296, −1.09106748077117273132267123094, −0.07788758608182126775474019558,
1.97071169620690148836275291670, 2.77499070373767609343670566404, 4.25496770989928235106067590059, 5.93447969545712093325307369918, 6.56282990414251101439329981487, 7.30509998548584824908319135223, 8.067028823780431373214839942975, 9.702928093843032410075275461223, 10.56000383446039921147236778948, 11.07331681222054533332218894910