| L(s) = 1 | + (−0.870 + 0.705i)2-s + (−0.448 − 1.67i)3-s + (−0.154 + 0.728i)4-s + (0.686 + 2.12i)5-s + (1.57 + 1.14i)6-s + (−3.87 − 1.03i)7-s + (−1.39 − 2.73i)8-s + (−2.59 + 1.50i)9-s + (−2.09 − 1.36i)10-s + (2.90 + 0.305i)11-s + (1.28 − 0.0679i)12-s + (−3.55 + 4.38i)13-s + (4.10 − 1.82i)14-s + (3.25 − 2.10i)15-s + (1.78 + 0.795i)16-s + (−2.38 + 1.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.615 + 0.498i)2-s + (−0.259 − 0.965i)3-s + (−0.0774 + 0.364i)4-s + (0.307 + 0.951i)5-s + (0.641 + 0.465i)6-s + (−1.46 − 0.392i)7-s + (−0.493 − 0.968i)8-s + (−0.865 + 0.500i)9-s + (−0.663 − 0.432i)10-s + (0.875 + 0.0919i)11-s + (0.371 − 0.0196i)12-s + (−0.985 + 1.21i)13-s + (1.09 − 0.488i)14-s + (0.839 − 0.543i)15-s + (0.446 + 0.198i)16-s + (−0.577 + 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0358145 + 0.264925i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0358145 + 0.264925i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.448 + 1.67i)T \) |
| 5 | \( 1 + (-0.686 - 2.12i)T \) |
| good | 2 | \( 1 + (0.870 - 0.705i)T + (0.415 - 1.95i)T^{2} \) |
| 7 | \( 1 + (3.87 + 1.03i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.90 - 0.305i)T + (10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (3.55 - 4.38i)T + (-2.70 - 12.7i)T^{2} \) |
| 17 | \( 1 + (2.38 - 1.21i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (6.58 + 2.14i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.283 - 0.737i)T + (-17.0 - 15.3i)T^{2} \) |
| 29 | \( 1 + (0.999 - 1.11i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (-3.68 - 4.08i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.780 + 4.92i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (2.76 - 0.290i)T + (40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-0.0852 + 0.318i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-7.80 + 0.409i)T + (46.7 - 4.91i)T^{2} \) |
| 53 | \( 1 + (-8.38 - 4.27i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (0.0903 + 0.859i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-1.20 + 11.5i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (7.23 + 0.378i)T + (66.6 + 7.00i)T^{2} \) |
| 71 | \( 1 + (0.558 - 0.181i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.29 - 8.20i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.976 - 0.879i)T + (8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (2.54 - 3.91i)T + (-33.7 - 75.8i)T^{2} \) |
| 89 | \( 1 + (2.06 + 1.50i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.530 - 10.1i)T + (-96.4 + 10.1i)T^{2} \) |
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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
−12.67547334973214433777215472256, −11.90350424631519789248926164923, −10.66425284367803073226477864131, −9.549918866702052413365507392513, −8.781440725118047862656661708569, −7.28355931555971564545086507208, −6.72618708543778029636441890000, −6.33977753602214098395402287047, −3.93770521092615020055609251491, −2.47220994024821539969780576085,
0.25886202776513221560554958325, 2.64071443575143163473886832124, 4.30069879606454273739574238706, 5.56733956686628100257438961484, 6.29899192196541639434207306219, 8.491007734992225118229153915963, 9.153846206263166837262393351858, 9.914070127169711882698266889847, 10.38351427054641528209181185125, 11.75761090973116954286780898781
