| L(s) = 1 | + (−1.08 + 0.908i)2-s + (0.881 − 0.471i)3-s + (0.349 − 1.96i)4-s + (1.51 + 1.84i)5-s + (−0.527 + 1.31i)6-s + (1.69 − 1.13i)7-s + (1.41 + 2.45i)8-s + (0.555 − 0.831i)9-s + (−3.31 − 0.623i)10-s + (1.92 − 0.585i)11-s + (−0.620 − 1.90i)12-s + (−1.76 − 1.44i)13-s + (−0.809 + 2.77i)14-s + (2.20 + 0.912i)15-s + (−3.75 − 1.37i)16-s + (−1.20 + 0.497i)17-s + ⋯ |
| L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.509 − 0.272i)3-s + (0.174 − 0.984i)4-s + (0.676 + 0.824i)5-s + (−0.215 + 0.535i)6-s + (0.641 − 0.428i)7-s + (0.498 + 0.866i)8-s + (0.185 − 0.277i)9-s + (−1.04 − 0.197i)10-s + (0.581 − 0.176i)11-s + (−0.179 − 0.548i)12-s + (−0.488 − 0.401i)13-s + (−0.216 + 0.740i)14-s + (0.568 + 0.235i)15-s + (−0.939 − 0.343i)16-s + (−0.291 + 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29679 + 0.341297i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29679 + 0.341297i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.08 - 0.908i)T \) |
| 3 | \( 1 + (-0.881 + 0.471i)T \) |
| good | 5 | \( 1 + (-1.51 - 1.84i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.69 + 1.13i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-1.92 + 0.585i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (1.76 + 1.44i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (1.20 - 0.497i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.0663 - 0.674i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-4.83 + 0.962i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (1.28 - 4.23i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-3.94 - 3.94i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.45 - 0.142i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (1.24 + 6.25i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (3.63 + 1.94i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-3.99 - 9.64i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.42 - 4.70i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (6.73 - 5.52i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (3.92 + 7.33i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-2.13 - 4.00i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (8.27 + 12.3i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (6.01 + 4.02i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.42 + 5.85i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-4.16 + 0.409i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (10.0 + 2.00i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (9.11 + 9.11i)T + 97iT^{2} \) |
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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
−10.92520572451154792807093349213, −10.47993662226547525162508363219, −9.427171034747286371743001050984, −8.643853726747585331095823774330, −7.59237242254582334927247543761, −6.89656447252691753048375116774, −6.00490625267053477440643553331, −4.69591440042243274562814112047, −2.85287051553552300000232685140, −1.46903793636015915168353248998,
1.48345669559854063054370597819, 2.58117435809163873297646277199, 4.16890328982567681333491604410, 5.16128707621665696781467200875, 6.75667935636142361170451516771, 7.940662310780939395228253269064, 8.795020550191700814754991516281, 9.376732415765089984485261301555, 10.05149091346186008132947851139, 11.32894834101758905661544787336
