| L(s) = 1 | + (−1.01 + 0.987i)2-s + (1.39 + 1.01i)3-s + (0.0499 − 1.99i)4-s + (0.122 − 2.23i)5-s + (−2.40 + 0.350i)6-s + 4.42i·7-s + (1.92 + 2.07i)8-s + (−0.0123 − 0.0380i)9-s + (2.08 + 2.38i)10-s + (5.07 + 1.65i)11-s + (2.09 − 2.73i)12-s + (−0.237 − 0.730i)13-s + (−4.36 − 4.47i)14-s + (2.42 − 2.98i)15-s + (−3.99 − 0.199i)16-s + (3.34 + 4.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.715 + 0.698i)2-s + (0.803 + 0.583i)3-s + (0.0249 − 0.999i)4-s + (0.0547 − 0.998i)5-s + (−0.982 + 0.143i)6-s + 1.67i·7-s + (0.680 + 0.733i)8-s + (−0.00412 − 0.0126i)9-s + (0.657 + 0.753i)10-s + (1.53 + 0.497i)11-s + (0.603 − 0.788i)12-s + (−0.0658 − 0.202i)13-s + (−1.16 − 1.19i)14-s + (0.626 − 0.770i)15-s + (−0.998 − 0.0498i)16-s + (0.811 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.928021 + 0.637566i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.928021 + 0.637566i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.01 - 0.987i)T \) |
| 5 | \( 1 + (-0.122 + 2.23i)T \) |
| good | 3 | \( 1 + (-1.39 - 1.01i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 - 4.42iT - 7T^{2} \) |
| 11 | \( 1 + (-5.07 - 1.65i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.237 + 0.730i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-3.34 - 4.60i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.01 + 1.40i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (5.01 + 1.62i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.21 + 1.66i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.10 - 1.53i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.01 + 6.18i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.754 + 2.32i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 6.65T + 43T^{2} \) |
| 47 | \( 1 + (-4.72 + 6.50i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.77 - 2.74i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.02 + 0.334i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (6.86 + 2.23i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-5.12 + 3.72i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.472 - 0.343i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (5.48 + 1.78i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-6.27 - 4.55i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (5.49 - 3.99i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.89 + 11.9i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.90 - 2.62i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−12.42909901183797364273321555113, −11.86311039620568915798354172711, −10.11845777816714508938988514425, −9.253075643396897689590520260848, −8.818364450408907859284100712849, −8.113803261534006656654449283262, −6.35853287788757568411917914908, −5.45127247391863620656323196182, −4.03931804442795935110281005608, −1.93407760111750138053766017262,
1.45609997685936741319756683600, 3.11195379071664407375725096739, 3.98959990113381640975462491444, 6.68169439042521151803032124639, 7.35760188116272581517093595694, 8.158122565216617002912317043153, 9.468394859553498944059065365657, 10.27110820185757367298022099895, 11.21344736083641877711133748383, 12.01487695767972844265564608535
