L(s) = 1 | + (1.71 + 1.71i)3-s − 2.46i·5-s + (−0.707 − 0.707i)7-s + 2.85i·9-s + (−3.89 − 3.89i)11-s + (−2.36 − 2.36i)13-s + (4.22 − 4.22i)15-s + (1.41 − 1.41i)17-s + (2.77 − 2.77i)19-s − 2.42i·21-s − 6.33·23-s − 1.09·25-s + (0.244 − 0.244i)27-s + (−1.38 − 1.38i)29-s + 6.75·31-s + ⋯ |
L(s) = 1 | + (0.988 + 0.988i)3-s − 1.10i·5-s + (−0.267 − 0.267i)7-s + 0.952i·9-s + (−1.17 − 1.17i)11-s + (−0.654 − 0.654i)13-s + (1.09 − 1.09i)15-s + (0.344 − 0.344i)17-s + (0.635 − 0.635i)19-s − 0.528i·21-s − 1.32·23-s − 0.219·25-s + (0.0470 − 0.0470i)27-s + (−0.256 − 0.256i)29-s + 1.21·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.698220484\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.698220484\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 41 | \( 1 + (-4.55 - 4.49i)T \) |
good | 3 | \( 1 + (-1.71 - 1.71i)T + 3iT^{2} \) |
| 5 | \( 1 + 2.46iT - 5T^{2} \) |
| 11 | \( 1 + (3.89 + 3.89i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.36 + 2.36i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + (-2.77 + 2.77i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.33T + 23T^{2} \) |
| 29 | \( 1 + (1.38 + 1.38i)T + 29iT^{2} \) |
| 31 | \( 1 - 6.75T + 31T^{2} \) |
| 37 | \( 1 - 0.731T + 37T^{2} \) |
| 43 | \( 1 + 7.12iT - 43T^{2} \) |
| 47 | \( 1 + (5.61 - 5.61i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.60 - 4.60i)T + 53iT^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 9.57iT - 61T^{2} \) |
| 67 | \( 1 + (3.13 - 3.13i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.44 + 2.44i)T + 71iT^{2} \) |
| 73 | \( 1 - 13.6iT - 73T^{2} \) |
| 79 | \( 1 + (1.22 + 1.22i)T + 79iT^{2} \) |
| 83 | \( 1 - 6.98T + 83T^{2} \) |
| 89 | \( 1 + (-11.5 - 11.5i)T + 89iT^{2} \) |
| 97 | \( 1 + (-11.0 + 11.0i)T - 97iT^{2} \) |
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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
−9.666776536649850513330671869321, −8.837938878371781617996367430066, −8.148172639042024623885308482546, −7.63021050516150560056025826503, −6.01842439572065138284085576455, −5.12093241748490206295577898066, −4.44977655109039681698448162495, −3.31528500302430706625628520797, −2.63113267452774179949638856583, −0.62200827317055214798882982822,
1.89635269311481446099248893466, 2.52959994068204849953040952603, 3.39230059045728618880132811982, 4.74783731695989142230730908524, 6.04203317866124966956892430350, 6.87198748636944717125003427374, 7.63423495908475593628994112973, 7.918306342008794460029186697988, 9.120792142086362236291197603536, 10.03888435728165638600392487757