L(s) = 1 | − 2-s + (−1 − 1.41i)3-s + 4-s − 1.41i·5-s + (1 + 1.41i)6-s − 4·7-s − 8-s + (−1.00 + 2.82i)9-s + 1.41i·10-s − 5.65i·11-s + (−1 − 1.41i)12-s − 4.24i·13-s + 4·14-s + (−2.00 + 1.41i)15-s + 16-s + 2.82i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.577 − 0.816i)3-s + 0.5·4-s − 0.632i·5-s + (0.408 + 0.577i)6-s − 1.51·7-s − 0.353·8-s + (−0.333 + 0.942i)9-s + 0.447i·10-s − 1.70i·11-s + (−0.288 − 0.408i)12-s − 1.17i·13-s + 1.06·14-s + (−0.516 + 0.365i)15-s + 0.250·16-s + 0.685i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.187891 - 0.416840i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.187891 - 0.416840i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 19 | \( 1 + (-1 - 4.24i)T \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4.24iT - 31T^{2} \) |
| 37 | \( 1 + 4.24iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 1.41iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 4.24iT - 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 8.48iT - 97T^{2} \) |
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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
−12.86554969930770695368190381367, −12.39727882625190788016740595340, −11.01841174469738338022426905919, −10.13004918113045919660344999085, −8.732450706857503879431768376690, −7.88739778826632931859306112493, −6.36644239338775995376605019153, −5.72842745695832435114139827872, −3.13805947770897882212007565114, −0.66816671282063161139195356884,
2.97323078393205506761813742798, 4.67072515898436690904446070940, 6.56900575356071618745082935569, 6.99016355562247833280236354883, 9.206044813673965073257274287356, 9.687257480288723321972873330012, 10.56265684308273267318571520182, 11.72666334402644106659205396212, 12.59996735577287581826995145004, 14.17461496914506874884294786783