| L(s) = 1 | + (−0.372 − 1.36i)2-s + (2.26 + 1.05i)3-s + (−1.72 + 1.01i)4-s + (2.33 + 3.33i)5-s + (0.595 − 3.48i)6-s + (−1.68 − 2.91i)7-s + (2.02 + 1.96i)8-s + (2.08 + 2.47i)9-s + (3.67 − 4.42i)10-s + (−4.07 + 1.09i)11-s + (−4.97 + 0.484i)12-s + (2.73 − 1.27i)13-s + (−3.35 + 3.38i)14-s + (1.76 + 10.0i)15-s + (1.93 − 3.50i)16-s + (2.69 + 2.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.263 − 0.964i)2-s + (1.30 + 0.609i)3-s + (−0.861 + 0.508i)4-s + (1.04 + 1.49i)5-s + (0.243 − 1.42i)6-s + (−0.636 − 1.10i)7-s + (0.717 + 0.696i)8-s + (0.693 + 0.826i)9-s + (1.16 − 1.39i)10-s + (−1.22 + 0.329i)11-s + (−1.43 + 0.139i)12-s + (0.757 − 0.353i)13-s + (−0.896 + 0.905i)14-s + (0.455 + 2.58i)15-s + (0.482 − 0.875i)16-s + (0.654 + 0.548i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70415 + 0.00632355i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70415 + 0.00632355i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.372 + 1.36i)T \) |
| 19 | \( 1 + (-4.35 - 0.247i)T \) |
| good | 3 | \( 1 + (-2.26 - 1.05i)T + (1.92 + 2.29i)T^{2} \) |
| 5 | \( 1 + (-2.33 - 3.33i)T + (-1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (1.68 + 2.91i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.07 - 1.09i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-2.73 + 1.27i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (-2.69 - 2.26i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.469 - 2.66i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.336 + 3.84i)T + (-28.5 - 5.03i)T^{2} \) |
| 31 | \( 1 + (4.63 + 8.03i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.99 + 5.99i)T + 37iT^{2} \) |
| 41 | \( 1 + (2.68 - 0.977i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.52 - 1.76i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (1.33 + 1.59i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-2.79 + 3.98i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (-4.09 + 0.358i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (-3.30 + 4.71i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (4.41 + 0.385i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (1.87 + 0.330i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.56 - 4.30i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-4.15 + 1.51i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (10.9 + 2.94i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (2.98 + 1.08i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (1.19 - 1.42i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.26312549130850074504255533035, −10.30212984663240048593274685027, −10.09816424735470334678653304616, −9.388809651388424880396484828718, −7.997997102406258954650571420547, −7.26504229410778986892835244469, −5.61676335815125896215517870374, −3.71726141948308693722152004898, −3.23772391962126524478543202822, −2.16309887822484270754220772348,
1.49370472154688983409055183417, 3.04995874731426196528817043702, 5.13481321194054575941696008405, 5.66836842051415263733966582880, 6.99183402179951803837123994505, 8.272628967674302406264001391322, 8.731757541868933859229072274001, 9.275131630746135887176679894817, 10.17779113259039762881623330504, 12.33901428299609588149803606960
