L(s) = 1 | + (−0.171 + 0.297i)5-s + (0.271 − 2.63i)7-s + (−2.45 − 4.26i)11-s + (0.974 + 1.68i)13-s + (2.07 − 3.59i)17-s + (0.202 + 0.350i)19-s + (−1.37 + 2.38i)23-s + (2.44 + 4.22i)25-s + (1.63 − 2.83i)29-s − 3.28·31-s + (0.736 + 0.533i)35-s + (−3.38 − 5.87i)37-s + (−2.81 − 4.87i)41-s + (−4.96 + 8.59i)43-s − 13.2·47-s + ⋯ |
L(s) = 1 | + (−0.0768 + 0.133i)5-s + (0.102 − 0.994i)7-s + (−0.741 − 1.28i)11-s + (0.270 + 0.467i)13-s + (0.502 − 0.871i)17-s + (0.0464 + 0.0803i)19-s + (−0.287 + 0.498i)23-s + (0.488 + 0.845i)25-s + (0.304 − 0.527i)29-s − 0.590·31-s + (0.124 + 0.0901i)35-s + (−0.557 − 0.965i)37-s + (−0.439 − 0.760i)41-s + (−0.756 + 1.31i)43-s − 1.92·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9589297042\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9589297042\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.271 + 2.63i)T \) |
good | 5 | \( 1 + (0.171 - 0.297i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.45 + 4.26i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.974 - 1.68i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.07 + 3.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.202 - 0.350i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.37 - 2.38i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.63 + 2.83i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 + (3.38 + 5.87i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.81 + 4.87i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.96 - 8.59i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 + (-1.38 + 2.39i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 8.45T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 8.42T + 67T^{2} \) |
| 71 | \( 1 + 5.08T + 71T^{2} \) |
| 73 | \( 1 + (-4.58 + 7.93i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 4.49T + 79T^{2} \) |
| 83 | \( 1 + (-4.62 + 8.01i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.54 - 6.13i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.31 - 2.27i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.649225965904160632390500746183, −7.87847623222072558589720031790, −7.28056483021865537561891239553, −6.43933366611606058724304520352, −5.51837047449110305643504223998, −4.76882769499729274506808984826, −3.61844140081075753417896966224, −3.09315082220949492825234717024, −1.59587492995262420837975655905, −0.32246822671699974405707735166,
1.59360421550282167780746859983, 2.53325939335677398884838150885, 3.51562486547542633071371860346, 4.75952564011464279536954063375, 5.22041075596101945581166580200, 6.21090676030834864755144376930, 6.94751345287691935098583810989, 8.070625533131041752722658275139, 8.345179115473992439128643049389, 9.290381379851510391703318863641