| L(s) = 1 | + (0.618 − 1.61i)3-s − 2.23·5-s − 5.23·7-s + (−2.23 − 2.00i)9-s + (−1.38 + 3.61i)15-s + (−3.23 + 8.47i)21-s + 5.70i·23-s + 5.00·25-s + (−4.61 + 2.38i)27-s + 6·29-s + 11.7·35-s − 12i·41-s + 11.2i·43-s + (5.00 + 4.47i)45-s − 13.7i·47-s + ⋯ |
| L(s) = 1 | + (0.356 − 0.934i)3-s − 0.999·5-s − 1.97·7-s + (−0.745 − 0.666i)9-s + (−0.356 + 0.934i)15-s + (−0.706 + 1.84i)21-s + 1.19i·23-s + 1.00·25-s + (−0.888 + 0.458i)27-s + 1.11·29-s + 1.97·35-s − 1.87i·41-s + 1.71i·43-s + (0.745 + 0.666i)45-s − 1.99i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6920851087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6920851087\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 + 1.61i)T \) |
| 5 | \( 1 + 2.23T \) |
| good | 7 | \( 1 + 5.23T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.70iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 12iT - 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 13.7iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 - 8.18iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 4.29T + 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.115365591252721180381405811777, −8.472325207204626714324109068252, −7.52714737963841010679996005428, −6.95704430847424740246822373561, −6.36251399546116178535706637469, −5.43591226437249346217464617614, −3.91808875249130013207707724221, −3.34431769883903162803898894090, −2.52506338656078348156080828548, −0.830386226316714201500292802031,
0.33406559510895863014558709123, 2.74350528394313331854383477052, 3.23747753694142801973158741297, 4.09567646597640512895996247776, 4.82499018857933670360990128283, 6.09605387708813105993272224185, 6.69026544663871064171687413278, 7.69608557192722666010814993555, 8.540204661585326246720798867002, 9.174790468012657029806259320400
