L(s) = 1 | + 1.04·3-s + (−0.980 − 3.01i)5-s + (0.809 − 0.587i)7-s − 1.91·9-s + (−0.720 + 2.21i)11-s + (−4.96 − 3.60i)13-s + (−1.02 − 3.14i)15-s + (−0.584 + 1.79i)17-s + (0.874 − 0.635i)19-s + (0.842 − 0.612i)21-s + (−0.0492 − 0.0357i)23-s + (−4.10 + 2.98i)25-s − 5.11·27-s + (−2.10 − 6.46i)29-s + (−2.72 + 8.38i)31-s + ⋯ |
L(s) = 1 | + 0.601·3-s + (−0.438 − 1.34i)5-s + (0.305 − 0.222i)7-s − 0.638·9-s + (−0.217 + 0.668i)11-s + (−1.37 − 1.00i)13-s + (−0.263 − 0.811i)15-s + (−0.141 + 0.436i)17-s + (0.200 − 0.145i)19-s + (0.183 − 0.133i)21-s + (−0.0102 − 0.00745i)23-s + (−0.821 + 0.596i)25-s − 0.985·27-s + (−0.390 − 1.20i)29-s + (−0.489 + 1.50i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7802705795\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7802705795\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 41 | \( 1 + (0.446 - 6.38i)T \) |
good | 3 | \( 1 - 1.04T + 3T^{2} \) |
| 5 | \( 1 + (0.980 + 3.01i)T + (-4.04 + 2.93i)T^{2} \) |
| 11 | \( 1 + (0.720 - 2.21i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (4.96 + 3.60i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.584 - 1.79i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.874 + 0.635i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.0492 + 0.0357i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.10 + 6.46i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.72 - 8.38i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (3.72 + 11.4i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-5.49 - 3.99i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-1.60 - 1.16i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.92 + 5.91i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.52 - 2.56i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (8.06 - 5.85i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.38 + 7.33i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.81 + 11.7i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + 9.13T + 73T^{2} \) |
| 79 | \( 1 + 7.10T + 79T^{2} \) |
| 83 | \( 1 - 2.63T + 83T^{2} \) |
| 89 | \( 1 + (-4.99 + 3.63i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (3.62 + 11.1i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.251156742449059050258281977912, −8.621599763936224425299606659237, −7.79175854890040105750202537207, −7.40101242574307305614170281179, −5.78904648826977483695389909204, −5.00828534153798466271321136498, −4.29968561388253387447264975352, −3.07719139901091909716518040008, −1.91195231517373308694612193768, −0.28860244704323867739412889540,
2.21256306763270184464477881433, 2.91310829762626528220236208056, 3.79419781522715638474999898424, 5.04481780462670323327071607286, 6.06513567477918799929044970978, 7.13412428755527163479350253167, 7.54084447998498623156179368837, 8.562811012422336680662687103366, 9.277019957899242284029360472192, 10.16813810773691895605326107740