| L(s) = 1 | + (1.02 + 2.40i)2-s + (−2.61 + 0.507i)3-s + (−3.36 + 3.48i)4-s + (−2.13 − 0.651i)5-s + (−3.89 − 5.77i)6-s + (−1.09 + 4.07i)7-s + (−6.95 − 2.66i)8-s + (3.78 − 1.53i)9-s + (−0.616 − 5.81i)10-s + (3.20 + 0.680i)11-s + (7.02 − 10.8i)12-s + (1.12 − 0.137i)13-s + (−10.9 + 1.53i)14-s + (5.92 + 0.616i)15-s + (−0.341 − 9.78i)16-s + (1.62 − 2.70i)17-s + ⋯ |
| L(s) = 1 | + (0.722 + 1.70i)2-s + (−1.50 + 0.293i)3-s + (−1.68 + 1.74i)4-s + (−0.956 − 0.291i)5-s + (−1.59 − 2.35i)6-s + (−0.413 + 1.54i)7-s + (−2.45 − 0.943i)8-s + (1.26 − 0.510i)9-s + (−0.195 − 1.83i)10-s + (0.965 + 0.205i)11-s + (2.02 − 3.12i)12-s + (0.310 − 0.0381i)13-s + (−2.92 + 0.410i)14-s + (1.52 + 0.159i)15-s + (−0.0854 − 2.44i)16-s + (0.394 − 0.656i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.332818 - 0.140074i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.332818 - 0.140074i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.13 + 0.651i)T \) |
| 19 | \( 1 + (4.07 - 1.53i)T \) |
| good | 2 | \( 1 + (-1.02 - 2.40i)T + (-1.38 + 1.43i)T^{2} \) |
| 3 | \( 1 + (2.61 - 0.507i)T + (2.78 - 1.12i)T^{2} \) |
| 7 | \( 1 + (1.09 - 4.07i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.20 - 0.680i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.12 + 0.137i)T + (12.6 - 3.14i)T^{2} \) |
| 17 | \( 1 + (-1.62 + 2.70i)T + (-7.98 - 15.0i)T^{2} \) |
| 23 | \( 1 + (1.04 - 3.41i)T + (-19.0 - 12.8i)T^{2} \) |
| 29 | \( 1 + (-2.20 + 8.83i)T + (-25.6 - 13.6i)T^{2} \) |
| 31 | \( 1 + (6.77 + 0.712i)T + (30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-2.75 - 5.41i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (9.56 - 0.334i)T + (40.9 - 2.86i)T^{2} \) |
| 43 | \( 1 + (1.00 + 1.44i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (6.64 - 3.99i)T + (22.0 - 41.4i)T^{2} \) |
| 53 | \( 1 + (-0.0581 - 3.32i)T + (-52.9 + 1.84i)T^{2} \) |
| 59 | \( 1 + (-6.62 + 4.13i)T + (25.8 - 53.0i)T^{2} \) |
| 61 | \( 1 + (-4.79 + 2.54i)T + (34.1 - 50.5i)T^{2} \) |
| 67 | \( 1 + (5.86 - 6.74i)T + (-9.32 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-3.09 + 1.50i)T + (43.7 - 55.9i)T^{2} \) |
| 73 | \( 1 + (6.29 + 0.772i)T + (70.8 + 17.6i)T^{2} \) |
| 79 | \( 1 + (-8.93 + 13.2i)T + (-29.5 - 73.2i)T^{2} \) |
| 83 | \( 1 + (-3.82 - 4.72i)T + (-17.2 + 81.1i)T^{2} \) |
| 89 | \( 1 + (0.327 - 9.37i)T + (-88.7 - 6.20i)T^{2} \) |
| 97 | \( 1 + (2.11 - 1.83i)T + (13.4 - 96.0i)T^{2} \) |
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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
−11.94138253045593839422202037262, −11.45185958700471000533213997034, −9.727575648818143284251701101983, −8.804884885801214384815559913382, −7.952386163003216364728339149924, −6.74306023848630981984671821116, −6.14916830894182302177261486210, −5.40510837242211301241200080168, −4.61172134283580879902397233126, −3.62316849451802216919675094851,
0.23691140083390665118034103522, 1.34448051255277354500645664210, 3.52191733714591152756752304446, 4.06737298979177468288999553400, 5.02546852454033674009382807130, 6.37308651714176037021605092588, 7.03369731166892820117102565402, 8.694514533886520366120942280386, 10.15574852194581138620918424084, 10.68348547555997494576794679476
