L(s) = 1 | + (−0.848 + 0.848i)3-s + (2.61 + 0.406i)7-s + 1.56i·9-s − 2.56·11-s + (2.17 − 2.17i)13-s + (2.17 + 2.17i)17-s + 8.54·19-s + (−2.56 + 1.87i)21-s + (−5.03 − 5.03i)23-s + (−3.86 − 3.86i)27-s + 5.68i·29-s + 4.79i·31-s + (2.17 − 2.17i)33-s + (2.82 − 2.82i)37-s + 3.68i·39-s + ⋯ |
L(s) = 1 | + (−0.489 + 0.489i)3-s + (0.988 + 0.153i)7-s + 0.520i·9-s − 0.772·11-s + (0.602 − 0.602i)13-s + (0.526 + 0.526i)17-s + 1.95·19-s + (−0.558 + 0.408i)21-s + (−1.05 − 1.05i)23-s + (−0.744 − 0.744i)27-s + 1.05i·29-s + 0.861i·31-s + (0.378 − 0.378i)33-s + (0.464 − 0.464i)37-s + 0.590i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.547723410\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.547723410\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.61 - 0.406i)T \) |
good | 3 | \( 1 + (0.848 - 0.848i)T - 3iT^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + (-2.17 + 2.17i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.17 - 2.17i)T + 17iT^{2} \) |
| 19 | \( 1 - 8.54T + 19T^{2} \) |
| 23 | \( 1 + (5.03 + 5.03i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.68iT - 29T^{2} \) |
| 31 | \( 1 - 4.79iT - 31T^{2} \) |
| 37 | \( 1 + (-2.82 + 2.82i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.54iT - 41T^{2} \) |
| 43 | \( 1 + (-0.620 - 0.620i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.848 + 0.848i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.41 - 4.41i)T + 53iT^{2} \) |
| 59 | \( 1 - 3.74T + 59T^{2} \) |
| 61 | \( 1 + 4.79iT - 61T^{2} \) |
| 67 | \( 1 + (-2.20 + 2.20i)T - 67iT^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + (-3.39 + 3.39i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.80iT - 79T^{2} \) |
| 83 | \( 1 + (5.66 - 5.66i)T - 83iT^{2} \) |
| 89 | \( 1 - 4.79T + 89T^{2} \) |
| 97 | \( 1 + (-13.3 - 13.3i)T + 97iT^{2} \) |
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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
−10.00306782682446647706940548683, −8.834002403497768200526621153791, −7.949952224759735128301278065360, −7.62546867471279546124277209207, −6.19695688666289878144933936507, −5.29458261538611341773577065550, −4.98095533293282904057691919666, −3.77904650060175060233385749847, −2.61087767303305474500626195700, −1.23787827953158241306384569773,
0.792433260004610187260843592829, 1.89850315216386053525017881185, 3.32279872542014680762109580136, 4.34252901071628583644879121634, 5.52612162176147017272741989806, 5.85677801352543645101745391232, 7.24480548503848804381099282633, 7.55458938222457679143480855367, 8.490275795004604603167695343807, 9.530674008690474204630321960130