L(s) = 1 | + (−0.841 − 2.31i)2-s + (2.14 + 0.779i)3-s + (−3.10 + 2.60i)4-s + (−2.84 + 0.500i)5-s − 5.60i·6-s + (0.251 + 1.42i)7-s + (4.36 + 2.51i)8-s + (1.67 + 1.40i)9-s + (3.54 + 6.14i)10-s + (1.43 − 2.48i)11-s + (−8.66 + 3.15i)12-s + (−0.863 − 1.02i)13-s + (3.08 − 1.78i)14-s + (−6.47 − 1.14i)15-s + (0.745 − 4.22i)16-s + (0.409 − 0.487i)17-s + ⋯ |
L(s) = 1 | + (−0.594 − 1.63i)2-s + (1.23 + 0.449i)3-s + (−1.55 + 1.30i)4-s + (−1.27 + 0.224i)5-s − 2.28i·6-s + (0.0950 + 0.539i)7-s + (1.54 + 0.890i)8-s + (0.558 + 0.468i)9-s + (1.12 + 1.94i)10-s + (0.432 − 0.748i)11-s + (−2.50 + 0.910i)12-s + (−0.239 − 0.285i)13-s + (0.824 − 0.475i)14-s + (−1.67 − 0.294i)15-s + (0.186 − 1.05i)16-s + (0.0992 − 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.515047 - 0.407073i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.515047 - 0.407073i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-4.38 + 4.21i)T \) |
good | 2 | \( 1 + (0.841 + 2.31i)T + (-1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (-2.14 - 0.779i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (2.84 - 0.500i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.251 - 1.42i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.43 + 2.48i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.863 + 1.02i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.409 + 0.487i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.636 + 1.74i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (6.38 - 3.68i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-8.44 - 4.87i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.28iT - 31T^{2} \) |
| 41 | \( 1 + (4.07 - 3.41i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 - 5.16iT - 43T^{2} \) |
| 47 | \( 1 + (5.53 + 9.58i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.46 - 8.29i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-5.35 - 0.943i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.559 - 0.666i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.975 - 5.52i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.480 - 0.174i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 + (-3.22 + 0.567i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (3.83 + 3.21i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.860 + 0.151i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (11.4 - 6.61i)T + (48.5 - 84.0i)T^{2} \) |
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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
−16.05774833134528136371328122720, −14.93238513626759305798210224800, −13.67520784095224996506847614552, −12.12510212001073026718347764705, −11.36820405805270742729210686279, −9.952848858741226680606297007987, −8.784797970141332334546817190012, −8.011075825309524472780970170857, −3.96117605352766742286988566884, −2.90585490194562999260867628957,
4.30588116630336647595851982164, 6.79596319845813941279063758426, 7.87245643330148126168482106347, 8.387921644080676364064047430893, 9.825874390548534431515993580238, 12.15182590990421579860157330151, 13.85180665798958664938494142632, 14.58362165979706031185909461216, 15.51990608913006003487821403235, 16.46425682905192028090476563264