| L(s) = 1 | + (1.26 + 0.642i)2-s + (0.611 + 3.86i)3-s + (1.17 + 1.61i)4-s + (−4.99 + 0.279i)5-s + (−1.70 + 5.26i)6-s + (−0.357 + 2.25i)7-s + (0.442 + 2.79i)8-s + (−5.99 + 1.94i)9-s + (−6.47 − 2.85i)10-s + (9.86 − 4.86i)11-s + (−5.53 + 5.53i)12-s + (−0.116 + 0.228i)13-s + (−1.89 + 2.61i)14-s + (−4.13 − 19.1i)15-s + (−1.23 + 3.80i)16-s + (6.66 + 13.0i)17-s + ⋯ |
| L(s) = 1 | + (0.630 + 0.321i)2-s + (0.203 + 1.28i)3-s + (0.293 + 0.404i)4-s + (−0.998 + 0.0559i)5-s + (−0.284 + 0.876i)6-s + (−0.0510 + 0.322i)7-s + (0.0553 + 0.349i)8-s + (−0.665 + 0.216i)9-s + (−0.647 − 0.285i)10-s + (0.896 − 0.442i)11-s + (−0.460 + 0.460i)12-s + (−0.00893 + 0.0175i)13-s + (−0.135 + 0.186i)14-s + (−0.275 − 1.27i)15-s + (−0.0772 + 0.237i)16-s + (0.392 + 0.769i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.991775 + 1.47343i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991775 + 1.47343i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 - 0.642i)T \) |
| 5 | \( 1 + (4.99 - 0.279i)T \) |
| 11 | \( 1 + (-9.86 + 4.86i)T \) |
| good | 3 | \( 1 + (-0.611 - 3.86i)T + (-8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (0.357 - 2.25i)T + (-46.6 - 15.1i)T^{2} \) |
| 13 | \( 1 + (0.116 - 0.228i)T + (-99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (-6.66 - 13.0i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-7.82 + 10.7i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-9.77 - 9.77i)T + 529iT^{2} \) |
| 29 | \( 1 + (-16.5 - 22.7i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (16.7 + 51.6i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-7.08 + 44.7i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (58.6 + 42.5i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (13.2 + 13.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (32.9 - 5.22i)T + (2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-4.76 + 9.34i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (21.8 + 30.0i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (13.5 - 41.5i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-82.2 + 82.2i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + (13.1 - 40.4i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-31.2 - 4.95i)T + (5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (98.9 - 32.1i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (91.2 - 46.5i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 - 110. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-128. - 65.3i)T + (5.53e3 + 7.61e3i)T^{2} \) |
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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
−14.09229929478781655790185079103, −12.66297344400540076981706342958, −11.61166170990007145730085981171, −10.78385250374944484061707295194, −9.374056977197691229004485016046, −8.416280289805832703850156599806, −7.01170740202474667531285838563, −5.44277649589355532432362801848, −4.15460379388572721472893232065, −3.37563265717749962181072929685,
1.25305328801379239912399960580, 3.20583343221216960955276789618, 4.70228899614509502945446768122, 6.59175675236752152690846354646, 7.31173962396104430557660358531, 8.451719285449157572501875535732, 10.08055749236999520950885531581, 11.62118380279669342571261685197, 12.07545884989531023390189406374, 12.98987209913526403161704123847
