| L(s) = 1 | + (0.492 − 0.456i)2-s + (−0.115 + 1.54i)4-s + (0.137 − 0.349i)5-s + (0.0148 + 2.64i)7-s + (1.48 + 1.86i)8-s + (−0.0921 − 0.234i)10-s + (−1.00 − 0.310i)11-s + (0.160 + 0.704i)13-s + (1.21 + 1.29i)14-s + (−1.47 − 0.223i)16-s + (−4.62 + 3.15i)17-s + (0.636 + 1.10i)19-s + (0.523 + 0.252i)20-s + (−0.637 + 0.306i)22-s + (3.63 + 2.47i)23-s + ⋯ |
| L(s) = 1 | + (0.348 − 0.323i)2-s + (−0.0578 + 0.772i)4-s + (0.0613 − 0.156i)5-s + (0.00560 + 0.999i)7-s + (0.525 + 0.658i)8-s + (−0.0291 − 0.0742i)10-s + (−0.303 − 0.0935i)11-s + (0.0446 + 0.195i)13-s + (0.325 + 0.346i)14-s + (−0.369 − 0.0557i)16-s + (−1.12 + 0.764i)17-s + (0.146 + 0.252i)19-s + (0.117 + 0.0564i)20-s + (−0.135 + 0.0654i)22-s + (0.757 + 0.516i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32859 + 0.815626i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32859 + 0.815626i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.0148 - 2.64i)T \) |
| good | 2 | \( 1 + (-0.492 + 0.456i)T + (0.149 - 1.99i)T^{2} \) |
| 5 | \( 1 + (-0.137 + 0.349i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (1.00 + 0.310i)T + (9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.160 - 0.704i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (4.62 - 3.15i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-0.636 - 1.10i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.63 - 2.47i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-7.21 - 3.47i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-3.98 + 6.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.742 + 9.91i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (0.563 + 0.706i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-0.746 + 0.936i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-1.43 + 1.32i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (0.322 - 4.30i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (4.99 + 12.7i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-0.317 - 4.24i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (1.21 - 2.10i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.99 + 2.40i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (1.58 + 1.47i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-3.65 - 6.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.36 + 14.7i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-6.67 + 2.05i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 9.74T + 97T^{2} \) |
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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
−11.38124694824629914030298644503, −10.67544675370511490858451008518, −9.223740546327368635365965393560, −8.666991758366389302033881690918, −7.74125292805246728177266736891, −6.57616605294571992064235283210, −5.38973731053617963342930190213, −4.41500218420809500338041501047, −3.17213570055345631793837844311, −2.10401952525621336761752656433,
0.929036311871520893751937803522, 2.81528958446052631134190261359, 4.48105322559577340616599033653, 4.97318537252916766809751228428, 6.56361608386595832558490266843, 6.81703732069030218235332029279, 8.138507414178040707735991932141, 9.264534342765892935212043976526, 10.34806037310652815179212697126, 10.64226971963097444773421779084
