| L(s) = 1 | + (−1.35 + 0.392i)2-s + (0.346 − 0.242i)3-s + (1.69 − 1.06i)4-s + (0.747 − 0.0653i)5-s + (−0.375 + 0.465i)6-s + (−2.51 + 1.45i)7-s + (−1.88 + 2.11i)8-s + (−0.964 + 2.65i)9-s + (−0.989 + 0.381i)10-s + (1.05 + 3.95i)11-s + (0.327 − 0.779i)12-s + (0.801 − 1.14i)13-s + (2.85 − 2.96i)14-s + (0.242 − 0.203i)15-s + (1.72 − 3.60i)16-s + (4.89 − 1.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.960 + 0.277i)2-s + (0.199 − 0.140i)3-s + (0.846 − 0.532i)4-s + (0.334 − 0.0292i)5-s + (−0.153 + 0.190i)6-s + (−0.951 + 0.549i)7-s + (−0.665 + 0.746i)8-s + (−0.321 + 0.883i)9-s + (−0.312 + 0.120i)10-s + (0.319 + 1.19i)11-s + (0.0945 − 0.225i)12-s + (0.222 − 0.317i)13-s + (0.762 − 0.791i)14-s + (0.0627 − 0.0526i)15-s + (0.431 − 0.901i)16-s + (1.18 − 0.432i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.642354 + 0.513733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.642354 + 0.513733i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 - 0.392i)T \) |
| 19 | \( 1 + (-1.35 - 4.14i)T \) |
| good | 3 | \( 1 + (-0.346 + 0.242i)T + (1.02 - 2.81i)T^{2} \) |
| 5 | \( 1 + (-0.747 + 0.0653i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (2.51 - 1.45i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.05 - 3.95i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.801 + 1.14i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-4.89 + 1.78i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-1.50 - 1.79i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.09 - 4.49i)T + (-18.6 + 22.2i)T^{2} \) |
| 31 | \( 1 + (0.887 + 1.53i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.19 + 4.19i)T - 37iT^{2} \) |
| 41 | \( 1 + (-3.04 - 0.537i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (7.32 - 0.641i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-3.47 - 1.26i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.910 + 10.4i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (4.92 - 10.5i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (10.3 + 0.906i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (4.83 + 10.3i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-7.21 + 8.59i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-11.7 - 2.07i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.578 - 3.28i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (1.14 - 4.25i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-10.3 + 1.81i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-11.6 + 4.24i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.91399752647061814561430255061, −10.68208461753841389576758550541, −9.766392998398417516090166931121, −9.313273989808723945594961652349, −8.033486738486219179829533025566, −7.33836368585953489968005870032, −6.12916731133438268399180840156, −5.28027612701643833420552164119, −3.11987769231290292633791488585, −1.80584588267175881447396562192,
0.826698769610236170737959458147, 2.94835989653181351352605414237, 3.75027256052860511263512412120, 6.02570059924991045315553017511, 6.60495603918275822164672185879, 7.896004329944397022517803954700, 8.924878032146116287612960508532, 9.587485244976815589444179715624, 10.36523386047168517557567090498, 11.40489229642670202607750636006
