L(s) = 1 | + (−0.951 − 0.309i)2-s + (−1.02 + 1.40i)3-s + (0.809 + 0.587i)4-s + (−0.814 − 2.08i)5-s + (1.40 − 1.02i)6-s − i·7-s + (−0.587 − 0.809i)8-s + (−0.00786 − 0.0241i)9-s + (0.131 + 2.23i)10-s + (−1.63 + 5.03i)11-s + (−1.65 + 0.537i)12-s + (0.332 − 0.108i)13-s + (−0.309 + 0.951i)14-s + (3.76 + 0.982i)15-s + (0.309 + 0.951i)16-s + (−2.50 − 3.44i)17-s + ⋯ |
L(s) = 1 | + (−0.672 − 0.218i)2-s + (−0.590 + 0.812i)3-s + (0.404 + 0.293i)4-s + (−0.364 − 0.931i)5-s + (0.574 − 0.417i)6-s − 0.377i·7-s + (−0.207 − 0.286i)8-s + (−0.00262 − 0.00806i)9-s + (0.0414 + 0.705i)10-s + (−0.493 + 1.51i)11-s + (−0.477 + 0.155i)12-s + (0.0921 − 0.0299i)13-s + (−0.0825 + 0.254i)14-s + (0.971 + 0.253i)15-s + (0.0772 + 0.237i)16-s + (−0.606 − 0.834i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0208719 + 0.159880i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0208719 + 0.159880i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 5 | \( 1 + (0.814 + 2.08i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (1.02 - 1.40i)T + (-0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (1.63 - 5.03i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.332 + 0.108i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (2.50 + 3.44i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (5.03 - 3.66i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (5.55 + 1.80i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (5.76 + 4.18i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.81 + 1.31i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.20 + 0.390i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.55 - 7.85i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 8.33iT - 43T^{2} \) |
| 47 | \( 1 + (-2.00 + 2.76i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (6.50 - 8.94i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.779 - 2.39i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.55 + 10.9i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-7.40 - 10.1i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (12.4 + 9.04i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.10 + 2.63i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-13.2 - 9.59i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (2.93 + 4.03i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.753 + 2.32i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.793 + 1.09i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−11.70210569194524016109456557731, −10.87542615324310049530548305886, −9.942649338010418429131388072662, −9.498819256026047915439471095182, −8.162287994586518402620933040530, −7.50025357353432437947132851247, −6.04359081833388640527473172129, −4.66622846394843187693543347515, −4.19568600691477133111484549177, −2.02832959357964041303514121074,
0.13891186449894179164269523965, 2.15943641093415326441481741164, 3.66896589371713309395988664214, 5.72502217812876666019644347179, 6.33402574710283193684668065931, 7.17412499771208136892952445792, 8.186514386064832427984407209482, 8.977835165157857007250216706758, 10.43868379307861819860442595022, 11.04646599144940123880242608621