| L(s) = 1 | + (−1.55 − 1.92i)2-s + (0.978 − 0.207i)3-s + (−0.860 + 4.04i)4-s + (−3.59 − 0.569i)5-s + (−1.92 − 1.55i)6-s + (0.829 + 1.27i)7-s + (4.72 − 2.40i)8-s + (0.913 − 0.406i)9-s + (4.50 + 7.80i)10-s + (−0.479 + 3.28i)11-s + 4.14i·12-s + (3.57 − 0.454i)13-s + (1.16 − 3.58i)14-s + (−3.63 + 0.190i)15-s + (−4.44 − 1.97i)16-s + (0.0870 + 0.828i)17-s + ⋯ |
| L(s) = 1 | + (−1.10 − 1.36i)2-s + (0.564 − 0.120i)3-s + (−0.430 + 2.02i)4-s + (−1.60 − 0.254i)5-s + (−0.786 − 0.636i)6-s + (0.313 + 0.482i)7-s + (1.67 − 0.851i)8-s + (0.304 − 0.135i)9-s + (1.42 + 2.46i)10-s + (−0.144 + 0.989i)11-s + 1.19i·12-s + (0.992 − 0.126i)13-s + (0.311 − 0.959i)14-s + (−0.938 + 0.0491i)15-s + (−1.11 − 0.494i)16-s + (0.0211 + 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.647626 - 0.228338i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.647626 - 0.228338i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 11 | \( 1 + (0.479 - 3.28i)T \) |
| 13 | \( 1 + (-3.57 + 0.454i)T \) |
| good | 2 | \( 1 + (1.55 + 1.92i)T + (-0.415 + 1.95i)T^{2} \) |
| 5 | \( 1 + (3.59 + 0.569i)T + (4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (-0.829 - 1.27i)T + (-2.84 + 6.39i)T^{2} \) |
| 17 | \( 1 + (-0.0870 - 0.828i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (1.44 + 0.0759i)T + (18.8 + 1.98i)T^{2} \) |
| 23 | \( 1 + (-4.79 + 2.76i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.98 - 5.38i)T + (3.03 + 28.8i)T^{2} \) |
| 31 | \( 1 + (-0.420 - 2.65i)T + (-29.4 + 9.57i)T^{2} \) |
| 37 | \( 1 + (0.0287 + 0.549i)T + (-36.7 + 3.86i)T^{2} \) |
| 41 | \( 1 + (3.23 - 4.98i)T + (-16.6 - 37.4i)T^{2} \) |
| 43 | \( 1 + (-1.22 + 2.12i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.55 - 8.92i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-3.56 - 2.58i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-9.08 + 5.89i)T + (23.9 - 53.8i)T^{2} \) |
| 61 | \( 1 + (10.8 - 1.14i)T + (59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-1.46 + 5.48i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.825 + 1.01i)T + (-14.7 - 69.4i)T^{2} \) |
| 73 | \( 1 + (6.44 - 12.6i)T + (-42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (8.29 - 11.4i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.486 + 3.06i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-3.02 - 0.809i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-17.4 - 6.68i)T + (72.0 + 64.9i)T^{2} \) |
| 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
−11.00631533051792332831713158999, −10.33655380052199593726091375347, −9.055467924020840528548566522086, −8.542929960129186082159154319452, −7.934745203246494214828403640383, −6.96152089217981065426840863874, −4.68241454611030236099658031180, −3.70575887074960449080410457888, −2.69798344808100324491605847433, −1.19711123960628732490705065476,
0.74252126558797105528406476924, 3.37974951862481883863317096631, 4.48505763238005250104540127798, 5.94284657301312689907889614149, 7.07490687799024868657922441700, 7.65658157661645276370857216052, 8.491033284263119192783644983531, 8.819727712453781636677484842875, 10.22784084387148753974600832159, 10.97889545173365455290440292895
