L(s) = 1 | + (−1.40 − 0.165i)2-s + (1.03 + 1.38i)3-s + (1.94 + 0.463i)4-s + (1.15 + 0.310i)5-s + (−1.23 − 2.11i)6-s + (0.356 − 0.616i)7-s + (−2.65 − 0.972i)8-s + (−0.840 + 2.87i)9-s + (−1.57 − 0.627i)10-s + (−2.28 + 0.611i)11-s + (1.37 + 3.17i)12-s + (4.90 + 1.31i)13-s + (−0.601 + 0.807i)14-s + (0.773 + 1.92i)15-s + (3.57 + 1.80i)16-s − 0.863i·17-s + ⋯ |
L(s) = 1 | + (−0.993 − 0.116i)2-s + (0.599 + 0.800i)3-s + (0.972 + 0.231i)4-s + (0.517 + 0.138i)5-s + (−0.502 − 0.864i)6-s + (0.134 − 0.233i)7-s + (−0.939 − 0.343i)8-s + (−0.280 + 0.959i)9-s + (−0.498 − 0.198i)10-s + (−0.687 + 0.184i)11-s + (0.398 + 0.917i)12-s + (1.36 + 0.364i)13-s + (−0.160 + 0.215i)14-s + (0.199 + 0.497i)15-s + (0.892 + 0.450i)16-s − 0.209i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.872609 + 0.356306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.872609 + 0.356306i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.165i)T \) |
| 3 | \( 1 + (-1.03 - 1.38i)T \) |
good | 5 | \( 1 + (-1.15 - 0.310i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-0.356 + 0.616i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.28 - 0.611i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-4.90 - 1.31i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + 0.863iT - 17T^{2} \) |
| 19 | \( 1 + (0.539 + 0.539i)T + 19iT^{2} \) |
| 23 | \( 1 + (0.689 - 0.398i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-8.31 + 2.22i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (4.18 - 2.41i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.95 + 6.95i)T + 37iT^{2} \) |
| 41 | \( 1 + (3.17 + 5.49i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.22 + 12.0i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-1.31 + 2.27i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.87 + 8.87i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.40 - 12.7i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (0.146 + 0.548i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-1.84 + 6.89i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 3.03iT - 71T^{2} \) |
| 73 | \( 1 - 11.6iT - 73T^{2} \) |
| 79 | \( 1 + (0.841 + 0.486i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.99 + 11.1i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 4.35T + 89T^{2} \) |
| 97 | \( 1 + (2.89 - 5.01i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.43225981029895579142025943052, −11.90615205976484695402405072436, −10.62107703872480506936550258939, −10.29188781281688409343030006209, −9.032058866290851376994759901116, −8.369055307480080378035049827050, −7.08550601283593515599870878043, −5.60815715154860426832392318133, −3.74819633168144509696347433223, −2.20041533735773751947474382540,
1.52596933061496290350472427046, 3.05457870911800453054223513901, 5.73622813436928917870020721821, 6.66338766067698220738286345052, 8.054090003228239500663549922049, 8.549600091385199942102382275448, 9.700385973552209800570774182401, 10.79346183286225871754862414503, 11.91403937982697725678221951613, 13.03619623954444828168945297673