| L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.248 − 0.0729i)3-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (0.169 − 0.195i)6-s + (0.663 − 4.61i)7-s + (0.959 − 0.281i)8-s + (−2.46 − 1.58i)9-s + (0.142 + 0.989i)10-s + (−1.30 − 2.85i)11-s + (0.107 + 0.235i)12-s + (0.713 + 4.96i)13-s + (3.92 + 2.52i)14-s + (−0.248 + 0.0729i)15-s + (−0.142 + 0.989i)16-s + (4.20 − 4.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (−0.143 − 0.0421i)3-s + (−0.327 − 0.377i)4-s + (0.376 − 0.241i)5-s + (0.0692 − 0.0799i)6-s + (0.250 − 1.74i)7-s + (0.339 − 0.0996i)8-s + (−0.822 − 0.528i)9-s + (0.0450 + 0.313i)10-s + (−0.393 − 0.862i)11-s + (0.0310 + 0.0680i)12-s + (0.197 + 1.37i)13-s + (1.04 + 0.673i)14-s + (−0.0641 + 0.0188i)15-s + (−0.0355 + 0.247i)16-s + (1.02 − 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.920572 - 0.301608i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.920572 - 0.301608i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (0.267 + 4.78i)T \) |
| good | 3 | \( 1 + (0.248 + 0.0729i)T + (2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (-0.663 + 4.61i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (1.30 + 2.85i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.713 - 4.96i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-4.20 + 4.85i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-3.72 - 4.30i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (2.62 - 3.03i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (0.120 - 0.0354i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.702 - 0.451i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (9.63 - 6.19i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-6.90 - 2.02i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 5.35T + 47T^{2} \) |
| 53 | \( 1 + (-0.381 + 2.65i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.766 - 5.33i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (4.71 - 1.38i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-2.96 + 6.49i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-3.63 + 7.96i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-10.0 - 11.6i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.735 + 5.11i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (11.6 + 7.49i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-10.8 - 3.17i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (1.33 - 0.856i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.97815711364618602194060080185, −11.04629348208453325556215703704, −10.07097249168424673098181940052, −9.145328191835523580803222390485, −8.038918335289018308540068357935, −7.10164772151935286838747855938, −6.08670457839510595434265370936, −4.91743876310372149970875957567, −3.53641440057934263616608019498, −0.959431092413723117263295760788,
2.12647945187824381530635788468, 3.14346084231158294423724687611, 5.31543173280733426516314991974, 5.68987420275563319292035221981, 7.65573308851209563413638902954, 8.490210720210615555162171536169, 9.488922506989218110774865690912, 10.41540986135137143615441161019, 11.35522821029883919938446679140, 12.21285612298653640929676484273
