| L(s) = 1 | + (−0.870 + 0.492i)2-s + (−1.86 − 1.75i)3-s + (0.515 − 0.856i)4-s + (−2.78 − 0.102i)5-s + (2.49 + 0.606i)6-s + (0.618 + 3.16i)7-s + (−0.0275 + 0.999i)8-s + (0.229 + 3.56i)9-s + (2.47 − 1.28i)10-s + (−2.25 − 3.07i)11-s + (−2.46 + 0.697i)12-s + (4.96 + 2.12i)13-s + (−2.09 − 2.45i)14-s + (5.03 + 5.08i)15-s + (−0.467 − 0.883i)16-s + (−2.51 + 0.0461i)17-s + ⋯ |
| L(s) = 1 | + (−0.615 + 0.347i)2-s + (−1.07 − 1.01i)3-s + (0.257 − 0.428i)4-s + (−1.24 − 0.0458i)5-s + (1.01 + 0.247i)6-s + (0.233 + 1.19i)7-s + (−0.00974 + 0.353i)8-s + (0.0764 + 1.18i)9-s + (0.783 − 0.405i)10-s + (−0.680 − 0.926i)11-s + (−0.711 + 0.201i)12-s + (1.37 + 0.588i)13-s + (−0.560 − 0.655i)14-s + (1.29 + 1.31i)15-s + (−0.116 − 0.220i)16-s + (−0.609 + 0.0112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.402016 - 0.217629i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.402016 - 0.217629i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.870 - 0.492i)T \) |
| 19 | \( 1 + (2.20 - 3.76i)T \) |
| good | 3 | \( 1 + (1.86 + 1.75i)T + (0.192 + 2.99i)T^{2} \) |
| 5 | \( 1 + (2.78 + 0.102i)T + (4.98 + 0.367i)T^{2} \) |
| 7 | \( 1 + (-0.618 - 3.16i)T + (-6.48 + 2.63i)T^{2} \) |
| 11 | \( 1 + (2.25 + 3.07i)T + (-3.28 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-4.96 - 2.12i)T + (8.97 + 9.40i)T^{2} \) |
| 17 | \( 1 + (2.51 - 0.0461i)T + (16.9 - 0.624i)T^{2} \) |
| 23 | \( 1 + (-3.91 - 1.18i)T + (19.1 + 12.7i)T^{2} \) |
| 29 | \( 1 + (-0.0415 - 0.0335i)T + (6.08 + 28.3i)T^{2} \) |
| 31 | \( 1 + (-0.0447 - 0.119i)T + (-23.3 + 20.3i)T^{2} \) |
| 37 | \( 1 + (-4.37 + 9.97i)T + (-25.0 - 27.2i)T^{2} \) |
| 41 | \( 1 + (-0.397 - 0.0365i)T + (40.3 + 7.49i)T^{2} \) |
| 43 | \( 1 + (0.466 + 10.1i)T + (-42.8 + 3.94i)T^{2} \) |
| 47 | \( 1 + (-3.81 + 3.99i)T + (-2.15 - 46.9i)T^{2} \) |
| 53 | \( 1 + (2.14 + 7.32i)T + (-44.6 + 28.5i)T^{2} \) |
| 59 | \( 1 + (4.61 + 10.0i)T + (-38.3 + 44.8i)T^{2} \) |
| 61 | \( 1 + (-9.25 + 1.89i)T + (56.0 - 23.9i)T^{2} \) |
| 67 | \( 1 + (7.27 - 1.07i)T + (64.1 - 19.4i)T^{2} \) |
| 71 | \( 1 + (-6.76 - 1.38i)T + (65.2 + 27.9i)T^{2} \) |
| 73 | \( 1 + (-0.913 + 1.65i)T + (-38.7 - 61.8i)T^{2} \) |
| 79 | \( 1 + (11.4 - 7.33i)T + (33.0 - 71.7i)T^{2} \) |
| 83 | \( 1 + (0.875 - 6.31i)T + (-79.8 - 22.5i)T^{2} \) |
| 89 | \( 1 + (-2.92 - 5.28i)T + (-47.3 + 75.3i)T^{2} \) |
| 97 | \( 1 + (-14.3 - 2.12i)T + (92.8 + 28.1i)T^{2} \) |
| 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
−10.67755440421214139889253331476, −8.974575934465478059300467480735, −8.447442905187623015882458013759, −7.71169316530731079503254778227, −6.74855724245113410053217522143, −5.95628914986394265315755147587, −5.31343939069651305010052278396, −3.74886792007342059172986825817, −2.01855057180497271413434560343, −0.50675253486199216581292311496,
0.832718010251227659003668578959, 3.15769744918667076311308098267, 4.41782403280281839146551203185, 4.54117810399649221756266344922, 6.18207383666195107925420069247, 7.22831147721553140553885743048, 7.943060484796539440530694512880, 8.920491368939320542474542804558, 10.09869983084786367462237192700, 10.63949105275863701814519722087
