| L(s) = 1 | + (−1.58 + 0.707i)3-s − 2.23i·5-s − 3.16·7-s + (2.00 − 2.23i)9-s + (1.58 + 3.53i)15-s + (5.00 − 2.23i)21-s + 1.41i·23-s − 5.00·25-s + (−1.58 + 4.94i)27-s + 8.94i·29-s + 7.07i·35-s + 4.47i·41-s + 3.16·43-s + (−5.00 − 4.47i)45-s + 9.89i·47-s + ⋯ |
| L(s) = 1 | + (−0.912 + 0.408i)3-s − 0.999i·5-s − 1.19·7-s + (0.666 − 0.745i)9-s + (0.408 + 0.912i)15-s + (1.09 − 0.487i)21-s + 0.294i·23-s − 1.00·25-s + (−0.304 + 0.952i)27-s + 1.66i·29-s + 1.19i·35-s + 0.698i·41-s + 0.482·43-s + (−0.745 − 0.666i)45-s + 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.217562 + 0.335624i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.217562 + 0.335624i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.58 - 0.707i)T \) |
| 5 | \( 1 + 2.23iT \) |
| good | 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 8.94iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 - 9.89iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 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
−10.24372555957235433476412956474, −9.420980739202258730011084256062, −8.994411953270206301841338129882, −7.70935782018550340354300780261, −6.67820325299228761361893828520, −5.93117106612778817243163614463, −5.09878363056611197615117930428, −4.21307212008163279330335721508, −3.17329350584286058062177368218, −1.24063232035320407966779014538,
0.22678783103678452616293081335, 2.16657626560552036309416434825, 3.30440736705999964417640059286, 4.42571946507821101850518522930, 5.82567852065197921845949694788, 6.26697650815978769768547089403, 7.08168107121660260701278072678, 7.74695406365640856079227827984, 9.084207865725342273864217775035, 10.16768035713722582771704083168
