L(s) = 1 | + (−0.292 − 1.70i)3-s + (3 − 3i)7-s + (−2.82 + i)9-s + 1.41i·11-s + (−4.24 − 4.24i)17-s − 4i·19-s + (−5.99 − 4.24i)21-s + (2.82 − 2.82i)23-s + (2.53 + 4.53i)27-s + 1.41·29-s + 2·31-s + (2.41 − 0.414i)33-s + (2 − 2i)37-s + 5.65i·41-s + (−2 − 2i)43-s + ⋯ |
L(s) = 1 | + (−0.169 − 0.985i)3-s + (1.13 − 1.13i)7-s + (−0.942 + 0.333i)9-s + 0.426i·11-s + (−1.02 − 1.02i)17-s − 0.917i·19-s + (−1.30 − 0.925i)21-s + (0.589 − 0.589i)23-s + (0.487 + 0.872i)27-s + 0.262·29-s + 0.359·31-s + (0.420 − 0.0721i)33-s + (0.328 − 0.328i)37-s + 0.883i·41-s + (−0.304 − 0.304i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.395006616\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.395006616\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.292 + 1.70i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-3 + 3i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (4.24 + 4.24i)T + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + (-2 + 2i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (2 + 2i)T + 43iT^{2} \) |
| 47 | \( 1 + (5.65 + 5.65i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.48 - 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-4 + 4i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 + (3 + 3i)T + 73iT^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + (2.82 - 2.82i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 + (-13 + 13i)T - 97iT^{2} \) |
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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
−9.292240294339336693782406359400, −8.449625273560461429269348567815, −7.61780310662649722462491500672, −7.06829887624432967768639841045, −6.36195223009988681703948942208, −4.92568993109496884404957976731, −4.55563619792721296359389216392, −2.91608857919995405952627524945, −1.79541117466357769325734428489, −0.61355404920622851044851102891,
1.75324699816212045526266982568, 2.98479601770405182424978645371, 4.11467524151267280266728841009, 4.97159001585485841619855820413, 5.69671394688587711301125808722, 6.45808988721811816930781415782, 8.015520197236927409889957202088, 8.479618069707434351060621477267, 9.181914116916814835516695050350, 10.05902522573125798598781703481