| L(s) = 1 | − i·3-s + (−1 + 2i)5-s − 4i·7-s − 9-s − 4·11-s + (2 + i)15-s + 4i·17-s − 4·21-s + 4i·23-s + (−3 − 4i)25-s + i·27-s − 6·29-s − 4·31-s + 4i·33-s + (8 + 4i)35-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.447 + 0.894i)5-s − 1.51i·7-s − 0.333·9-s − 1.20·11-s + (0.516 + 0.258i)15-s + 0.970i·17-s − 0.872·21-s + 0.834i·23-s + (−0.600 − 0.800i)25-s + 0.192i·27-s − 1.11·29-s − 0.718·31-s + 0.696i·33-s + (1.35 + 0.676i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1 - 2i)T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−9.851059057813114294716576139893, −8.368259070688319206124819723465, −7.67839654102252744609709856331, −7.17609614322463757056987667011, −6.35418492804859390939354441409, −5.20457877827392035258724508329, −3.91926678170556330330360908835, −3.18905211706540382915751011087, −1.75010445711386402868872118949, 0,
2.16526019093913173537277476252, 3.20330954501237634905713369263, 4.53013155083203388550760594377, 5.29511194129143547094813093334, 5.80941734994912013695025701801, 7.33526250831731147424647497661, 8.203035708950935373983655346043, 8.956154360361886487202524769823, 9.404676613215999233946332963402
