| 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 | \( 1 + (-11.3 + 15.6i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−11.49922374887262647018512420772, −11.08339727715649368165187193345, −9.891208477795868614651526064310, −8.567120992462641066657579608755, −7.48493209579343170263652862370, −7.08268560164311447007715936520, −5.75845430444199383715612210612, −4.62939800400546910276287570368, −3.33446973892559559550564262973, −2.22266791318650001757417155022,
1.28547024768496973751061715664, 3.69876153643756198456151381090, 3.77855859789031543884307386327, 5.33142808004565180658451312072, 6.28086044168790673443083498380, 7.62197462072527155820159513170, 8.676102249140123545316086234211, 9.398168149905901356051580309512, 10.72912057591723077854543271521, 11.34774030254506340721516872364
