| L(s) = 1 | + (0.682 + 1.18i)2-s + (−0.498 − 2.82i)3-s + (0.0684 − 0.118i)4-s + (−3.58 − 1.30i)5-s + (3.00 − 2.51i)6-s + (4.26 + 1.55i)7-s + 2.91·8-s + (−4.91 + 1.79i)9-s + (−0.904 − 5.12i)10-s + (−0.180 + 1.02i)11-s + (−0.368 − 0.134i)12-s + (−1.07 − 0.391i)13-s + (1.07 + 6.10i)14-s + (−1.90 + 10.7i)15-s + (1.85 + 3.21i)16-s + (1.78 + 3.08i)17-s + ⋯ |
| L(s) = 1 | + (0.482 + 0.835i)2-s + (−0.287 − 1.63i)3-s + (0.0342 − 0.0592i)4-s + (−1.60 − 0.583i)5-s + (1.22 − 1.02i)6-s + (1.61 + 0.587i)7-s + 1.03·8-s + (−1.63 + 0.596i)9-s + (−0.286 − 1.62i)10-s + (−0.0544 + 0.308i)11-s + (−0.106 − 0.0387i)12-s + (−0.298 − 0.108i)13-s + (0.287 + 1.63i)14-s + (−0.490 + 2.78i)15-s + (0.463 + 0.802i)16-s + (0.431 + 0.748i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08509 - 0.388997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08509 - 0.388997i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (-10.2 + 2.00i)T \) |
| good | 2 | \( 1 + (-0.682 - 1.18i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.498 + 2.82i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (3.58 + 1.30i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-4.26 - 1.55i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (0.180 - 1.02i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (1.07 + 0.391i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.78 - 3.08i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.555 + 0.961i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.25 - 2.17i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0398 + 0.226i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.23 + 0.448i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (1.92 - 0.700i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 - 0.890T + 41T^{2} \) |
| 43 | \( 1 + (-1.09 - 1.89i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (9.24 + 7.75i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (3.66 + 1.33i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.81 - 10.3i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.209 + 0.175i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (12.5 - 4.55i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.69 + 9.86i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.75 + 9.93i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (9.29 - 3.38i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.75 - 15.6i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (1.96 - 1.65i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (9.34 + 3.40i)T + (74.3 + 62.3i)T^{2} \) |
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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
−13.60521338522314875384530868613, −12.44833144150587194274320091179, −11.83306546614317461204724923029, −11.01649403293855795370761638844, −8.303303410443617905604828372723, −7.899861186451588579134262356850, −7.07116047718415744333942260183, −5.62308588921591727428907283350, −4.57619198246832205379067618840, −1.57363205705691477477311010991,
3.24442101808397618680045799174, 4.25810067730486796430127964243, 4.84967059072390519228013107457, 7.44528922335001015300437329713, 8.317305744147583618979630615456, 10.15576427413335928809791572761, 11.11787053909305937757345725845, 11.28474945338915709935966639428, 12.19249622186932555190714453374, 14.16547274427181447971888094065
