| L(s) = 1 | + (1.96 + 1.96i)3-s + (0.254 + 2.22i)5-s + (−1.87 + 1.87i)7-s + 4.74i·9-s + (−4.88 − 4.88i)13-s + (−3.87 + 4.87i)15-s + 2.41i·19-s − 7.36·21-s + (6.74 + 6.74i)23-s + (−4.87 + 1.12i)25-s + (−3.42 + 3.42i)27-s + (−4.63 − 3.68i)35-s − 19.2i·39-s + (−10.5 + 1.20i)45-s − 7i·49-s + ⋯ |
| L(s) = 1 | + (1.13 + 1.13i)3-s + (0.113 + 0.993i)5-s + (−0.707 + 0.707i)7-s + 1.58i·9-s + (−1.35 − 1.35i)13-s + (−0.999 + 1.25i)15-s + 0.552i·19-s − 1.60·21-s + (1.40 + 1.40i)23-s + (−0.974 + 0.225i)25-s + (−0.659 + 0.659i)27-s + (−0.782 − 0.622i)35-s − 3.07i·39-s + (−1.57 + 0.179i)45-s − i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.866245961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.866245961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.254 - 2.22i)T \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
| good | 3 | \( 1 + (-1.96 - 1.96i)T + 3iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (4.88 + 4.88i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 - 2.41iT - 19T^{2} \) |
| 23 | \( 1 + (-6.74 - 6.74i)T + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 - 10.4iT - 59T^{2} \) |
| 61 | \( 1 - 6.47T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 - 8.25iT - 79T^{2} \) |
| 83 | \( 1 + (12.2 + 12.2i)T + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97iT^{2} \) |
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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
−9.932845137888075074756215341474, −9.601051815325812007330223117806, −8.724022066294817996449206203844, −7.77800974073797881833693891943, −7.07470428673175184516418723889, −5.77523916079108207849619256728, −5.03231879601749117464921558050, −3.65599229508767578457780372132, −3.04581784440595423325050525473, −2.41458152777243426972978050309,
0.68279097708874942884705672610, 1.96734492080600686819250883195, 2.87636288762870517391529274895, 4.15559894726787507964281741132, 5.01306648387652536023299991255, 6.62935354344275076644179636116, 6.94127369455650325828688386147, 7.82607386141163618640588624766, 8.706525653978998145264451821281, 9.274661241747621544947809741975
