| L(s) = 1 | + (0.951 − 0.309i)2-s + (0.454 + 0.625i)3-s + (0.809 − 0.587i)4-s + (−1.27 − 1.83i)5-s + (0.625 + 0.454i)6-s − i·7-s + (0.587 − 0.809i)8-s + (0.742 − 2.28i)9-s + (−1.77 − 1.35i)10-s + (−1.04 − 3.22i)11-s + (0.735 + 0.239i)12-s + (5.05 + 1.64i)13-s + (−0.309 − 0.951i)14-s + (0.571 − 1.63i)15-s + (0.309 − 0.951i)16-s + (−2.43 + 3.35i)17-s + ⋯ |
| L(s) = 1 | + (0.672 − 0.218i)2-s + (0.262 + 0.361i)3-s + (0.404 − 0.293i)4-s + (−0.569 − 0.821i)5-s + (0.255 + 0.185i)6-s − 0.377i·7-s + (0.207 − 0.286i)8-s + (0.247 − 0.761i)9-s + (−0.562 − 0.428i)10-s + (−0.315 − 0.971i)11-s + (0.212 + 0.0690i)12-s + (1.40 + 0.455i)13-s + (−0.0825 − 0.254i)14-s + (0.147 − 0.421i)15-s + (0.0772 − 0.237i)16-s + (−0.590 + 0.813i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.77888 - 0.886054i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77888 - 0.886054i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.27 + 1.83i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + (-0.454 - 0.625i)T + (-0.927 + 2.85i)T^{2} \) |
| 11 | \( 1 + (1.04 + 3.22i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-5.05 - 1.64i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.43 - 3.35i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-5.39 - 3.91i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.249 - 0.0810i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (7.94 - 5.77i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.74 + 2.72i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.340 + 0.110i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.254 + 0.782i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.32iT - 43T^{2} \) |
| 47 | \( 1 + (-2.75 - 3.79i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.624 - 0.859i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.489 - 1.50i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.462 - 1.42i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-4.68 + 6.45i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-6.82 + 4.95i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.37 - 0.448i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (5.16 - 3.75i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.26 - 4.49i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-4.39 - 13.5i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-11.3 - 15.6i)T + (-29.9 + 92.2i)T^{2} \) |
| 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
−11.34774030254506340721516872364, −10.72912057591723077854543271521, −9.398168149905901356051580309512, −8.676102249140123545316086234211, −7.62197462072527155820159513170, −6.28086044168790673443083498380, −5.33142808004565180658451312072, −3.77855859789031543884307386327, −3.69876153643756198456151381090, −1.28547024768496973751061715664,
2.22266791318650001757417155022, 3.33446973892559559550564262973, 4.62939800400546910276287570368, 5.75845430444199383715612210612, 7.08268560164311447007715936520, 7.48493209579343170263652862370, 8.567120992462641066657579608755, 9.891208477795868614651526064310, 11.08339727715649368165187193345, 11.49922374887262647018512420772
