| L(s) = 1 | + (−0.373 − 0.445i)2-s + (0.289 + 1.64i)3-s + (0.288 − 1.63i)4-s + (0.294 − 0.0519i)5-s + (0.622 − 0.741i)6-s + (2.63 + 0.241i)7-s + (−1.84 + 1.06i)8-s + (0.211 − 0.0769i)9-s + (−0.133 − 0.111i)10-s − 0.329·11-s + 2.76·12-s + (2.46 + 2.06i)13-s + (−0.876 − 1.26i)14-s + (0.170 + 0.468i)15-s + (−1.96 − 0.713i)16-s + (−0.871 + 2.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.264 − 0.314i)2-s + (0.167 + 0.947i)3-s + (0.144 − 0.818i)4-s + (0.131 − 0.0232i)5-s + (0.254 − 0.302i)6-s + (0.995 + 0.0911i)7-s + (−0.651 + 0.376i)8-s + (0.0704 − 0.0256i)9-s + (−0.0421 − 0.0353i)10-s − 0.0992·11-s + 0.799·12-s + (0.684 + 0.574i)13-s + (−0.234 − 0.337i)14-s + (0.0440 + 0.120i)15-s + (−0.490 − 0.178i)16-s + (−0.211 + 0.580i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11486 - 0.0488200i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11486 - 0.0488200i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.63 - 0.241i)T \) |
| 19 | \( 1 + (0.878 + 4.26i)T \) |
| good | 2 | \( 1 + (0.373 + 0.445i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.289 - 1.64i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-0.294 + 0.0519i)T + (4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + 0.329T + 11T^{2} \) |
| 13 | \( 1 + (-2.46 - 2.06i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.871 - 2.39i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (5.92 + 4.97i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (2.99 + 0.528i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (0.899 + 1.55i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.89 - 1.67i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.51 - 5.46i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.53 + 0.921i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.564 + 1.55i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-11.2 - 1.98i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (5.11 + 1.86i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.30 + 8.70i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.63 + 5.51i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.28 - 6.27i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.75 + 0.839i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.421 + 1.15i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (1.68 + 0.975i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.262 + 1.48i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.82 - 10.3i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−13.44999880919487157244592034347, −11.82526589438376516073191520739, −10.97900701287841942797440668886, −10.22433532427098927204097785904, −9.262101921408017243587440270611, −8.339940807490081881557628618263, −6.54014768970207285806214873914, −5.21458768628660489609480801765, −4.08566508613037794803738488411, −1.92668041874331732331267200187,
1.94988089118351894346681196884, 3.85354371744326186834575589254, 5.75604807831439544916545018088, 7.13796085214644061043059460465, 7.87856707200714009217676589221, 8.577013467255728226852399484937, 10.15503348712944128708992984282, 11.55150041174908654068075465930, 12.25908288824342777063437029086, 13.33536476699161561069809970780
