| L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.427 − 0.125i)3-s + (−0.654 − 0.755i)4-s + (−0.0300 + 0.0192i)5-s + (0.291 − 0.336i)6-s + (−0.142 + 0.989i)7-s + (0.959 − 0.281i)8-s + (−2.35 − 1.51i)9-s + (−0.00507 − 0.0353i)10-s + (2.36 + 5.16i)11-s + (0.185 + 0.405i)12-s + (0.862 + 6.00i)13-s + (−0.841 − 0.540i)14-s + (0.0152 − 0.00448i)15-s + (−0.142 + 0.989i)16-s + (−0.549 + 0.634i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (−0.246 − 0.0724i)3-s + (−0.327 − 0.377i)4-s + (−0.0134 + 0.00863i)5-s + (0.119 − 0.137i)6-s + (−0.0537 + 0.374i)7-s + (0.339 − 0.0996i)8-s + (−0.785 − 0.504i)9-s + (−0.00160 − 0.0111i)10-s + (0.711 + 1.55i)11-s + (0.0534 + 0.117i)12-s + (0.239 + 1.66i)13-s + (−0.224 − 0.144i)14-s + (0.00394 − 0.00115i)15-s + (−0.0355 + 0.247i)16-s + (−0.133 + 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.427776 + 0.714121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.427776 + 0.714121i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (2.15 + 4.28i)T \) |
| good | 3 | \( 1 + (0.427 + 0.125i)T + (2.52 + 1.62i)T^{2} \) |
| 5 | \( 1 + (0.0300 - 0.0192i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (-2.36 - 5.16i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.862 - 6.00i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.549 - 0.634i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-0.597 - 0.689i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (4.70 - 5.42i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-6.50 + 1.91i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-2.05 - 1.32i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-4.25 + 2.73i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (3.47 + 1.02i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 8.21T + 47T^{2} \) |
| 53 | \( 1 + (0.647 - 4.50i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.49 + 10.3i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-13.1 + 3.86i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-3.90 + 8.54i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.65 + 5.81i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-8.88 - 10.2i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.872 + 6.06i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-15.1 - 9.75i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (1.89 + 0.555i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-3.42 + 2.19i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.90153077277668037800210750655, −11.12196550506521715947960995581, −9.627477096002137302952630339609, −9.254919572794581666535614306785, −8.168111856803080561783864064957, −6.84502537065671456982374852377, −6.40295080114909771561310332655, −5.06817846939926849147142281115, −3.94285527214157857541280214191, −1.88798031865331245402266356613,
0.69402520232670017444430593790, 2.79190113018832198796161669318, 3.82109291574332850024557632314, 5.38157487801093657356122007748, 6.23456443815816433068824953757, 7.950800159320515931618674357773, 8.368963782029996253357812656445, 9.636050882829071050026254119174, 10.53995308199565240241223675259, 11.31788016574635116974384070634
