| L(s) = 1 | + (−0.399 + 0.916i)2-s + (1.60 − 0.651i)3-s + (−0.680 − 0.733i)4-s + (−1.48 − 0.223i)5-s + (−0.0446 + 1.73i)6-s + (−3.43 − 3.18i)7-s + (0.943 − 0.330i)8-s + (2.15 − 2.09i)9-s + (0.798 − 1.27i)10-s + (−1.19 + 1.02i)11-s + (−1.56 − 0.733i)12-s + (−2.74 + 4.02i)13-s + (4.29 − 1.87i)14-s + (−2.52 + 0.607i)15-s + (−0.0747 + 0.997i)16-s + (−3.43 − 3.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.282 + 0.648i)2-s + (0.926 − 0.376i)3-s + (−0.340 − 0.366i)4-s + (−0.663 − 0.0999i)5-s + (−0.0182 + 0.706i)6-s + (−1.29 − 1.20i)7-s + (0.333 − 0.116i)8-s + (0.717 − 0.697i)9-s + (0.252 − 0.401i)10-s + (−0.360 + 0.309i)11-s + (−0.452 − 0.211i)12-s + (−0.761 + 1.11i)13-s + (1.14 − 0.500i)14-s + (−0.652 + 0.156i)15-s + (−0.0186 + 0.249i)16-s + (−0.832 − 0.832i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.461 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.320661 - 0.528538i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.320661 - 0.528538i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.399 - 0.916i)T \) |
| 3 | \( 1 + (-1.60 + 0.651i)T \) |
| 29 | \( 1 + (-0.165 + 5.38i)T \) |
| good | 5 | \( 1 + (1.48 + 0.223i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (3.43 + 3.18i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (1.19 - 1.02i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.74 - 4.02i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (3.43 + 3.43i)T + 17iT^{2} \) |
| 19 | \( 1 + (6.10 + 3.83i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-7.09 - 2.78i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.509 - 0.375i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-0.418 - 1.19i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (1.78 + 0.478i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-4.44 + 6.02i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (-1.18 - 1.37i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-1.56 - 1.25i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (-9.90 + 5.71i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.97 + 0.0738i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (0.669 - 0.0502i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (3.54 - 1.70i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.851 + 7.55i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (2.28 - 12.0i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (-4.02 + 13.0i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (-8.03 + 0.905i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (5.43 + 10.2i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−10.25550406077975260253088769985, −9.425587385723376389497566683343, −8.866106267142552503137188870014, −7.63700041865180903958353285612, −7.01229788769076451465853508560, −6.60696079793237419961460344435, −4.60774449263234476649666669096, −3.89679209573029756050294603845, −2.47909827063414494032968510923, −0.33438326296944592824481285527,
2.38069571702161850247139948369, 3.09324078107948420282600544238, 4.05512915505280039191803119554, 5.42128495391157880355936217072, 6.75111928402461916189839719798, 7.994183479461342829627344450089, 8.637074943011442931731368693806, 9.317344744531971290539885850419, 10.30577518816736594935772281162, 10.83304658265244186740152626690
