L(s) = 1 | + (−1.22 − 0.698i)2-s + (1.02 + 1.71i)4-s + (−1.91 + 1.15i)5-s + 3.02i·7-s + (−0.0563 − 2.82i)8-s + (3.16 − 0.0813i)10-s + (1.30 − 1.30i)11-s + (−1.56 + 1.56i)13-s + (2.11 − 3.72i)14-s + (−1.90 + 3.51i)16-s + 2.98·17-s + (1.25 − 1.25i)19-s + (−3.94 − 2.10i)20-s + (−2.51 + 0.692i)22-s − 7.82·23-s + ⋯ |
L(s) = 1 | + (−0.869 − 0.494i)2-s + (0.511 + 0.859i)4-s + (−0.856 + 0.516i)5-s + 1.14i·7-s + (−0.0199 − 0.999i)8-s + (0.999 − 0.0257i)10-s + (0.393 − 0.393i)11-s + (−0.434 + 0.434i)13-s + (0.565 − 0.995i)14-s + (−0.476 + 0.878i)16-s + 0.724·17-s + (0.288 − 0.288i)19-s + (−0.881 − 0.471i)20-s + (−0.536 + 0.147i)22-s − 1.63·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0751899 + 0.284383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0751899 + 0.284383i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 + 0.698i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.91 - 1.15i)T \) |
good | 7 | \( 1 - 3.02iT - 7T^{2} \) |
| 11 | \( 1 + (-1.30 + 1.30i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.56 - 1.56i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.98T + 17T^{2} \) |
| 19 | \( 1 + (-1.25 + 1.25i)T - 19iT^{2} \) |
| 23 | \( 1 + 7.82T + 23T^{2} \) |
| 29 | \( 1 + (7.12 - 7.12i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.0502iT - 31T^{2} \) |
| 37 | \( 1 + (6.22 + 6.22i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 + (6.18 - 6.18i)T - 43iT^{2} \) |
| 47 | \( 1 + 5.87iT - 47T^{2} \) |
| 53 | \( 1 + (-4.40 + 4.40i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.20 - 8.20i)T - 59iT^{2} \) |
| 61 | \( 1 + (4.93 + 4.93i)T + 61iT^{2} \) |
| 67 | \( 1 + (1.29 + 1.29i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.9iT - 71T^{2} \) |
| 73 | \( 1 - 3.85T + 73T^{2} \) |
| 79 | \( 1 - 1.07iT - 79T^{2} \) |
| 83 | \( 1 + (7.16 - 7.16i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.60T + 89T^{2} \) |
| 97 | \( 1 + 9.70iT - 97T^{2} \) |
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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
−10.78196879227195058653997853314, −9.879354446953130630936541100572, −9.025628032857189743451940336236, −8.340712983391039708209586649707, −7.47936429306985344308692934376, −6.66014260724226028790771127465, −5.49068517441809568278688903870, −3.92882978736535712303733181532, −3.08390819259159421793345879324, −1.88486098131660318821065224464,
0.20165593541708231054156542160, 1.61121002710050973314464598685, 3.55984125004835822482496908270, 4.58532976387284237614071851555, 5.69125071382576404464429236469, 6.84358786126917243129911787674, 7.76197820380183249928394038011, 7.956256297431365621382463484457, 9.207086090538805916638352017536, 10.01685751870770814263585648656