L(s) = 1 | + (−0.939 + 0.342i)2-s + (1.00 − 2.77i)3-s + (0.766 − 0.642i)4-s + (0.557 − 2.16i)5-s + 2.94i·6-s + (−3.81 + 0.672i)7-s + (−0.500 + 0.866i)8-s + (−4.35 − 3.65i)9-s + (0.216 + 2.22i)10-s + (0.398 − 0.690i)11-s + (−1.00 − 2.77i)12-s + (2.06 − 1.73i)13-s + (3.35 − 1.93i)14-s + (−5.43 − 3.72i)15-s + (0.173 − 0.984i)16-s + (0.125 + 0.105i)17-s + ⋯ |
L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.582 − 1.59i)3-s + (0.383 − 0.321i)4-s + (0.249 − 0.968i)5-s + 1.20i·6-s + (−1.44 + 0.254i)7-s + (−0.176 + 0.306i)8-s + (−1.45 − 1.21i)9-s + (0.0684 + 0.703i)10-s + (0.120 − 0.208i)11-s + (−0.291 − 0.799i)12-s + (0.573 − 0.480i)13-s + (0.896 − 0.517i)14-s + (−1.40 − 0.962i)15-s + (0.0434 − 0.246i)16-s + (0.0303 + 0.0254i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.239289 - 0.897436i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.239289 - 0.897436i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 5 | \( 1 + (-0.557 + 2.16i)T \) |
| 37 | \( 1 + (2.54 + 5.52i)T \) |
good | 3 | \( 1 + (-1.00 + 2.77i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (3.81 - 0.672i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.398 + 0.690i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.06 + 1.73i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.125 - 0.105i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (1.41 - 3.87i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-3.01 - 5.22i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.297 - 0.171i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.59iT - 31T^{2} \) |
| 41 | \( 1 + (-8.50 + 7.13i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 - 9.32T + 43T^{2} \) |
| 47 | \( 1 + (6.52 - 3.76i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.98 + 1.23i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (4.82 + 0.851i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (5.93 + 7.07i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-8.06 + 1.42i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.42 - 1.97i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + 7.77iT - 73T^{2} \) |
| 79 | \( 1 + (-9.00 + 1.58i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (5.14 - 6.13i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-6.57 - 1.15i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (7.58 + 13.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.06461133759657073078368789062, −9.561886527681177237542871173397, −9.126101054653501983374229928870, −8.128836018926934762349799572826, −7.45740561516125721422170437392, −6.23231732434480553511203627129, −5.82344613010014334033819964743, −3.46573593508115498607748176508, −2.07835475253762972053167798359, −0.71545247470150207701588167865,
2.75089542417370954788983065552, 3.36585132364445832224221075062, 4.51806709463937298904799838381, 6.25160427576534734961784646190, 7.01343381097180849145936080910, 8.531552509473694703438037970750, 9.335393675460646096822278577160, 9.858992897122316337497125354161, 10.66387641731108746121544231760, 11.12566095290323590216719951117