| L(s) = 1 | + (1.06 + 0.226i)2-s + (−1.68 + 0.418i)3-s + (−0.747 − 0.332i)4-s + (−2.22 − 0.206i)5-s + (−1.88 + 0.0647i)6-s + (1.73 − 3.01i)7-s + (−2.47 − 1.80i)8-s + (2.65 − 1.40i)9-s + (−2.32 − 0.722i)10-s + (−3.44 − 0.731i)11-s + (1.39 + 0.246i)12-s + (−2.12 + 0.450i)13-s + (2.53 − 2.81i)14-s + (3.82 − 0.583i)15-s + (−1.13 − 1.25i)16-s + (−1.54 − 1.12i)17-s + ⋯ |
| L(s) = 1 | + (0.751 + 0.159i)2-s + (−0.970 + 0.241i)3-s + (−0.373 − 0.166i)4-s + (−0.995 − 0.0924i)5-s + (−0.768 + 0.0264i)6-s + (0.657 − 1.13i)7-s + (−0.876 − 0.636i)8-s + (0.883 − 0.468i)9-s + (−0.733 − 0.228i)10-s + (−1.03 − 0.220i)11-s + (0.402 + 0.0712i)12-s + (−0.588 + 0.125i)13-s + (0.676 − 0.751i)14-s + (0.988 − 0.150i)15-s + (−0.283 − 0.314i)16-s + (−0.375 − 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.270967 - 0.479554i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.270967 - 0.479554i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.68 - 0.418i)T \) |
| 5 | \( 1 + (2.22 + 0.206i)T \) |
| good | 2 | \( 1 + (-1.06 - 0.226i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (-1.73 + 3.01i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.44 + 0.731i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (2.12 - 0.450i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (1.54 + 1.12i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.970 + 0.704i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.34 + 1.49i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.374 - 3.55i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.663 - 6.31i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-2.46 + 7.58i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (2.53 - 0.538i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-5.34 + 9.26i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.37 + 13.0i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (11.2 - 8.18i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-5.32 + 1.13i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (0.516 + 0.109i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-1.12 + 10.7i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-6.38 + 4.63i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.75 + 8.46i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.706 + 6.72i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (3.25 - 1.44i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.71 - 8.36i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.79 - 17.0i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.08119972402473103741047967099, −10.90668057877616932940605961995, −10.44888300412439307292389055069, −8.997681923211853873339327967031, −7.61609339231269096684347509215, −6.76642787496624007981404131968, −5.19504896554096458370036737740, −4.67544417155993510063987851309, −3.66359021891960820591442068939, −0.40369458033868185430777028623,
2.62577591904910876637164850117, 4.39493361177751209673746532239, 5.07315412729407083945667263267, 6.08636903797519251583452739616, 7.67381219292137853623803340545, 8.349240085695696300454457776956, 9.803630032034808023418219885780, 11.33108472997677656930865504162, 11.55278194006925163097921509208, 12.71969567246239179900556437677
