| L(s) = 1 | − 0.537i·3-s + (−2.07 − 0.826i)5-s − 3.18i·7-s + 2.71·9-s − 4.15·11-s + 2.07i·13-s + (−0.443 + 1.11i)15-s − 5.79i·17-s − 19-s − 1.71·21-s + 2.60i·23-s + (3.63 + 3.43i)25-s − 3.06i·27-s − 6·29-s − 2.59·31-s + ⋯ |
| L(s) = 1 | − 0.310i·3-s + (−0.929 − 0.369i)5-s − 1.20i·7-s + 0.903·9-s − 1.25·11-s + 0.574i·13-s + (−0.114 + 0.288i)15-s − 1.40i·17-s − 0.229·19-s − 0.373·21-s + 0.543i·23-s + (0.726 + 0.686i)25-s − 0.590i·27-s − 1.11·29-s − 0.466·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3842658039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3842658039\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.07 + 0.826i)T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.537iT - 3T^{2} \) |
| 7 | \( 1 + 3.18iT - 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 - 2.07iT - 13T^{2} \) |
| 17 | \( 1 + 5.79iT - 17T^{2} \) |
| 23 | \( 1 - 2.60iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 2.59T + 31T^{2} \) |
| 37 | \( 1 + 4.30iT - 37T^{2} \) |
| 41 | \( 1 + 0.599T + 41T^{2} \) |
| 43 | \( 1 + 3.18iT - 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 11.7iT - 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 + 8.75T + 61T^{2} \) |
| 67 | \( 1 - 4.76iT - 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 2.72iT - 73T^{2} \) |
| 79 | \( 1 + 1.40T + 79T^{2} \) |
| 83 | \( 1 - 7.07iT - 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 2.07iT - 97T^{2} \) |
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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
−9.068972810720216837508058552250, −7.88294584280566883776204383468, −7.35153437686306517008449441922, −7.09589759313470583172300570115, −5.62993742511682734869543898016, −4.57243028722614835223907689665, −4.09952876691591596594841846598, −2.92064076815739047177013823143, −1.41654437170528093855821120845, −0.15303362611773562077013736034,
1.95983500701880417875286438867, 3.05706819607674734004801953736, 3.94507285876924600888578362166, 4.94228863464920909026787235515, 5.73056064158093050934180359985, 6.71972100826478792109785664968, 7.69859820250072258202066310837, 8.249025245421416970190821095416, 8.987911102440531551972259947322, 10.15710345438139353024610870546
