| L(s) = 1 | + (0.415 + 0.909i)2-s + (0.0141 − 0.00415i)3-s + (−0.654 + 0.755i)4-s + (−3.50 − 2.25i)5-s + (0.00966 + 0.0111i)6-s + (−0.142 − 0.989i)7-s + (−0.959 − 0.281i)8-s + (−2.52 + 1.62i)9-s + (0.592 − 4.12i)10-s + (−0.164 + 0.359i)11-s + (−0.00612 + 0.0134i)12-s + (0.524 − 3.64i)13-s + (0.841 − 0.540i)14-s + (−0.0589 − 0.0173i)15-s + (−0.142 − 0.989i)16-s + (−5.01 − 5.78i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (0.00817 − 0.00239i)3-s + (−0.327 + 0.377i)4-s + (−1.56 − 1.00i)5-s + (0.00394 + 0.00455i)6-s + (−0.0537 − 0.374i)7-s + (−0.339 − 0.0996i)8-s + (−0.841 + 0.540i)9-s + (0.187 − 1.30i)10-s + (−0.0495 + 0.108i)11-s + (−0.00176 + 0.00387i)12-s + (0.145 − 1.01i)13-s + (0.224 − 0.144i)14-s + (−0.0152 − 0.00447i)15-s + (−0.0355 − 0.247i)16-s + (−1.21 − 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.439 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.198171 - 0.317436i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.198171 - 0.317436i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.99 + 4.36i)T \) |
| good | 3 | \( 1 + (-0.0141 + 0.00415i)T + (2.52 - 1.62i)T^{2} \) |
| 5 | \( 1 + (3.50 + 2.25i)T + (2.07 + 4.54i)T^{2} \) |
| 11 | \( 1 + (0.164 - 0.359i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.524 + 3.64i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (5.01 + 5.78i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (4.06 - 4.68i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (0.216 + 0.250i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-6.68 - 1.96i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (3.66 - 2.35i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-3.16 - 2.03i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (3.79 - 1.11i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 4.21T + 47T^{2} \) |
| 53 | \( 1 + (-0.444 - 3.09i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.721 + 5.01i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (5.98 + 1.75i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (5.49 + 12.0i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (1.10 + 2.41i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-3.11 + 3.58i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.237 + 1.65i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-3.36 + 2.16i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (7.84 - 2.30i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-10.0 - 6.45i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.49852923738085014745081445477, −10.58719679551479834526458446483, −8.995646413807079932562142916734, −8.244961339763587174544094724052, −7.73949895962790048862904772465, −6.51084008575079498878517680953, −5.03917993968019712687576272915, −4.46019356021682187780886833881, −3.13450995092218791159592643186, −0.23379136441721633489323231350,
2.50874572591471271227834266230, 3.63206423472044965000021919890, 4.44651765024389072595040988465, 6.20031387196034446413766271431, 6.95838915398591138908775054666, 8.390575149480147018074116619850, 8.961377998550362228081480785862, 10.47576597541713790615032749082, 11.38620432804193612798331438749, 11.51365332137276824347188458399
