| L(s) = 1 | + (0.150 − 0.855i)2-s + (−1.85 + 0.674i)3-s + (1.17 + 0.426i)4-s + (0.633 − 0.230i)5-s + (0.297 + 1.68i)6-s + (2.63 + 0.237i)7-s + (1.40 − 2.44i)8-s + (0.679 − 0.569i)9-s + (−0.101 − 0.576i)10-s + 2.88·11-s − 2.45·12-s + (−0.0914 − 0.518i)13-s + (0.600 − 2.21i)14-s + (−1.01 + 0.853i)15-s + (0.0343 + 0.0287i)16-s + (−3.33 − 2.80i)17-s + ⋯ |
| L(s) = 1 | + (0.106 − 0.604i)2-s + (−1.06 + 0.389i)3-s + (0.585 + 0.213i)4-s + (0.283 − 0.103i)5-s + (0.121 + 0.688i)6-s + (0.995 + 0.0897i)7-s + (0.498 − 0.862i)8-s + (0.226 − 0.189i)9-s + (−0.0321 − 0.182i)10-s + 0.869·11-s − 0.709·12-s + (−0.0253 − 0.143i)13-s + (0.160 − 0.592i)14-s + (−0.262 + 0.220i)15-s + (0.00857 + 0.00719i)16-s + (−0.810 − 0.679i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07612 - 0.185829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07612 - 0.185829i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.63 - 0.237i)T \) |
| 19 | \( 1 + (2.19 - 3.76i)T \) |
| good | 2 | \( 1 + (-0.150 + 0.855i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (1.85 - 0.674i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.633 + 0.230i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 13 | \( 1 + (0.0914 + 0.518i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.33 + 2.80i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.216 - 1.22i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (7.70 + 2.80i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (2.69 - 4.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.19 + 3.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.375 - 2.13i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (1.95 + 1.63i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.94 + 2.46i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (11.6 + 4.24i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.51 + 1.27i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.60 + 9.08i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.876 - 4.97i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.299 - 0.251i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (9.01 - 3.28i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-9.96 - 8.36i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.46 - 2.53i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-15.0 - 5.48i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.48 + 1.99i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.86914508853147604821152377186, −11.76961366384750202718577361411, −11.33351924087449522974379584043, −10.56401428835005597790696946089, −9.332377931889869721309430590190, −7.77589063651200196531271777716, −6.44868466232042724499479156974, −5.28530076699364105946584790199, −3.96923747141170266048924858536, −1.86727859708128680509559885000,
1.82360694506266454098974248487, 4.59580634987824075753874572833, 5.86759179078079579544582621129, 6.54224322892790004301704756488, 7.60470001227834254159110546913, 8.979030459499420500921401469820, 10.73348671402378580562262423744, 11.25324183103333887330178798823, 12.04899772122466343857936060722, 13.38185991632373853096620873471
