| L(s) = 1 | + (−1.32 − 0.489i)2-s + (0.540 − 0.841i)3-s + (1.52 + 1.29i)4-s + (3.08 + 1.40i)5-s + (−1.12 + 0.851i)6-s + (2.20 + 0.648i)7-s + (−1.38 − 2.46i)8-s + (−0.415 − 0.909i)9-s + (−3.40 − 3.37i)10-s + (3.12 + 2.70i)11-s + (1.91 − 0.577i)12-s + (−1.13 − 3.86i)13-s + (−2.61 − 1.94i)14-s + (2.85 − 1.83i)15-s + (0.628 + 3.95i)16-s + (0.524 + 3.64i)17-s + ⋯ |
| L(s) = 1 | + (−0.938 − 0.345i)2-s + (0.312 − 0.485i)3-s + (0.760 + 0.649i)4-s + (1.37 + 0.629i)5-s + (−0.460 + 0.347i)6-s + (0.835 + 0.245i)7-s + (−0.489 − 0.872i)8-s + (−0.138 − 0.303i)9-s + (−1.07 − 1.06i)10-s + (0.941 + 0.815i)11-s + (0.552 − 0.166i)12-s + (−0.314 − 1.07i)13-s + (−0.698 − 0.519i)14-s + (0.736 − 0.473i)15-s + (0.157 + 0.987i)16-s + (0.127 + 0.884i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46909 - 0.130257i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46909 - 0.130257i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 + 0.489i)T \) |
| 3 | \( 1 + (-0.540 + 0.841i)T \) |
| 23 | \( 1 + (0.614 - 4.75i)T \) |
| good | 5 | \( 1 + (-3.08 - 1.40i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-2.20 - 0.648i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-3.12 - 2.70i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.13 + 3.86i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.524 - 3.64i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (2.78 + 0.400i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (9.62 - 1.38i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-6.32 + 4.06i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (9.92 - 4.53i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (1.35 - 2.96i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-6.12 + 9.53i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 6.59T + 47T^{2} \) |
| 53 | \( 1 + (-3.25 + 11.0i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (0.133 + 0.454i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (2.22 + 3.45i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-8.45 + 7.32i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (1.89 + 2.18i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.273 + 1.89i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (7.47 - 2.19i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.19 + 0.547i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-4.22 - 2.71i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.55 + 14.3i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.52079639697118344492450300250, −9.890890902950292170459015144595, −9.101945840370632332506756659430, −8.185539716369954001333620974042, −7.28903767552269800447591144206, −6.43802368961999694607500244842, −5.48591361812796625691653357533, −3.61490693563503472335104277818, −2.20043880985619493540418065208, −1.65675889160414191115592415232,
1.34159285795831206288890642133, 2.38946828361483374725647541003, 4.34922432012210768840052755941, 5.40528055964981979447547050552, 6.27037322967069076387122487007, 7.30266063558378122042535499739, 8.594770119327238815119015580372, 9.010790527563115818766825729935, 9.658097616812029733912627844002, 10.58742405277729680005700859074
