| L(s) = 1 | + (1.90 + 1.38i)2-s + (−0.104 − 0.994i)3-s + (1.09 + 3.37i)4-s + (−0.646 − 1.11i)5-s + (1.17 − 2.03i)6-s + (−3.84 − 0.816i)7-s + (−1.12 + 3.46i)8-s + (−0.978 + 0.207i)9-s + (0.318 − 3.02i)10-s + (−2.28 + 2.53i)11-s + (3.24 − 1.44i)12-s + (5.55 + 2.47i)13-s + (−6.18 − 6.87i)14-s + (−1.04 + 0.759i)15-s + (−1.20 + 0.875i)16-s + (−2.04 − 2.26i)17-s + ⋯ |
| L(s) = 1 | + (1.34 + 0.978i)2-s + (−0.0603 − 0.574i)3-s + (0.548 + 1.68i)4-s + (−0.289 − 0.500i)5-s + (0.480 − 0.832i)6-s + (−1.45 − 0.308i)7-s + (−0.398 + 1.22i)8-s + (−0.326 + 0.0693i)9-s + (0.100 − 0.957i)10-s + (−0.689 + 0.765i)11-s + (0.935 − 0.416i)12-s + (1.54 + 0.686i)13-s + (−1.65 − 1.83i)14-s + (−0.270 + 0.196i)15-s + (−0.301 + 0.218i)16-s + (−0.494 − 0.549i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49731 + 0.578969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49731 + 0.578969i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.104 + 0.994i)T \) |
| 31 | \( 1 + (-5.28 + 1.75i)T \) |
| good | 2 | \( 1 + (-1.90 - 1.38i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (0.646 + 1.11i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (3.84 + 0.816i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (2.28 - 2.53i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-5.55 - 2.47i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (2.04 + 2.26i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.80 + 1.24i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (1.03 - 3.17i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.98 + 2.16i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (2.33 - 4.03i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0333 + 0.316i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (8.75 - 3.89i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (9.57 - 6.95i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.53 + 0.538i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (1.06 + 10.1i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 + (5.39 + 9.34i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.02 - 0.642i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (0.405 - 0.450i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-8.41 - 9.34i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-1.09 + 10.4i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-5.00 - 15.3i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (2.73 + 8.41i)T + (-78.4 + 57.0i)T^{2} \) |
| 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
−13.74580269576609877247430711382, −13.36009127835376664837672758345, −12.59135628468206350732007216600, −11.53436158301580520462249035784, −9.668765071232008701001042184452, −8.111442370204699501923063626206, −6.89670975827840586676712947069, −6.19918978661718840044912667408, −4.72303231314322898960580947042, −3.33879380085004827241807078173,
3.07710301845630247217481305086, 3.65239391279867724051215619381, 5.47907612323004386430190487778, 6.38262471108986127686482988279, 8.609120030389534697663579973955, 10.23223603918709668977636219787, 10.78988211728966923660312090442, 11.84194523134465642041995657166, 13.08156762853768911165402287316, 13.48336240795311629322829538339
