L(s) = 1 | + (−1.79 − 1.79i)2-s + (−0.491 + 1.66i)3-s + 4.47i·4-s + (3.87 − 2.10i)6-s + (0.707 − 0.707i)7-s + (4.45 − 4.45i)8-s + (−2.51 − 1.63i)9-s − 1.56i·11-s + (−7.43 − 2.19i)12-s + (−2.21 − 2.21i)13-s − 2.54·14-s − 7.09·16-s + (3.60 + 3.60i)17-s + (1.59 + 7.46i)18-s + 1.68i·19-s + ⋯ |
L(s) = 1 | + (−1.27 − 1.27i)2-s + (−0.283 + 0.958i)3-s + 2.23i·4-s + (1.58 − 0.859i)6-s + (0.267 − 0.267i)7-s + (1.57 − 1.57i)8-s + (−0.839 − 0.543i)9-s − 0.472i·11-s + (−2.14 − 0.634i)12-s + (−0.615 − 0.615i)13-s − 0.680·14-s − 1.77·16-s + (0.874 + 0.874i)17-s + (0.375 + 1.75i)18-s + 0.385i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.619902 - 0.164697i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.619902 - 0.164697i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.491 - 1.66i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 + (1.79 + 1.79i)T + 2iT^{2} \) |
| 11 | \( 1 + 1.56iT - 11T^{2} \) |
| 13 | \( 1 + (2.21 + 2.21i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.60 - 3.60i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.68iT - 19T^{2} \) |
| 23 | \( 1 + (-0.995 + 0.995i)T - 23iT^{2} \) |
| 29 | \( 1 - 8.91T + 29T^{2} \) |
| 31 | \( 1 - 2.74T + 31T^{2} \) |
| 37 | \( 1 + (0.440 - 0.440i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.44iT - 41T^{2} \) |
| 43 | \( 1 + (-5.47 - 5.47i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.69 - 3.69i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.83 + 2.83i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.54T + 59T^{2} \) |
| 61 | \( 1 - 7.40T + 61T^{2} \) |
| 67 | \( 1 + (-3.75 + 3.75i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.61iT - 71T^{2} \) |
| 73 | \( 1 + (-5.89 - 5.89i)T + 73iT^{2} \) |
| 79 | \( 1 + 17.0iT - 79T^{2} \) |
| 83 | \( 1 + (-3.21 + 3.21i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.40T + 89T^{2} \) |
| 97 | \( 1 + (4.39 - 4.39i)T - 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.43813416878552193044249994225, −10.22526862982835275717895662836, −9.308170609823339898474246235257, −8.367091505589450882946507339518, −7.79838600591418476169755745578, −6.17657499580964066114691456398, −4.79476583010762708293791330343, −3.62610738137239562616302176282, −2.73248835275547646897003079619, −0.928592873889060889421495769307,
0.874964456220734838023374145107, 2.35077446527450845681041609929, 4.91760912830430285206552319687, 5.70439818221018128223873778460, 6.83232864300312333589839920272, 7.22651504580154973562547676906, 8.109027907110932073363327527419, 8.901507389186640192711916555151, 9.753523059154297700782126481308, 10.67024220694512061716712619674