| L(s) = 1 | + (0.164 − 0.758i)3-s + (0.970 − 2.01i)5-s + (1.35 − 2.48i)7-s + (2.18 + 0.996i)9-s + (1.93 − 1.67i)11-s + (−0.0157 + 0.00861i)13-s + (−1.36 − 1.06i)15-s + (3.80 + 5.08i)17-s + (0.577 + 4.01i)19-s + (−1.66 − 1.44i)21-s + (−2.37 − 4.16i)23-s + (−3.11 − 3.91i)25-s + (2.50 − 3.35i)27-s + (−8.22 − 1.18i)29-s + (0.792 + 0.509i)31-s + ⋯ |
| L(s) = 1 | + (0.0952 − 0.437i)3-s + (0.434 − 0.900i)5-s + (0.513 − 0.940i)7-s + (0.727 + 0.332i)9-s + (0.582 − 0.504i)11-s + (−0.00437 + 0.00238i)13-s + (−0.352 − 0.275i)15-s + (0.922 + 1.23i)17-s + (0.132 + 0.921i)19-s + (−0.362 − 0.314i)21-s + (−0.495 − 0.868i)23-s + (−0.623 − 0.782i)25-s + (0.482 − 0.645i)27-s + (−1.52 − 0.219i)29-s + (0.142 + 0.0915i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59884 - 1.25306i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59884 - 1.25306i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.970 + 2.01i)T \) |
| 23 | \( 1 + (2.37 + 4.16i)T \) |
| good | 3 | \( 1 + (-0.164 + 0.758i)T + (-2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-1.35 + 2.48i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-1.93 + 1.67i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.0157 - 0.00861i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-3.80 - 5.08i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.577 - 4.01i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (8.22 + 1.18i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.792 - 0.509i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.791 + 2.12i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-2.12 - 4.64i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.716 - 0.155i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (0.448 + 0.448i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.66 - 0.911i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-1.43 + 4.87i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (6.37 - 9.91i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (0.860 + 12.0i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-3.07 + 3.55i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.27 - 0.956i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (5.06 + 1.48i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (12.6 - 4.70i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (1.15 - 0.744i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.81 - 0.675i)T + (73.3 + 63.5i)T^{2} \) |
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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
−10.03343037190938720062303724761, −9.004933705927894185364883952996, −8.021030542107549285241373520229, −7.65539544820017715064077364715, −6.40160595806928517675591467777, −5.62801683474299310162457395707, −4.43224513841180775527935495023, −3.77355317853720848225221786844, −1.84905318937938924656518020199, −1.11074698237587694257396522950,
1.69347544757823890634519031338, 2.84684628199237861518157618114, 3.88855259632325960910256017500, 5.07898085643195340512084715119, 5.82539749022048901656724696891, 7.02642688294896751229649377670, 7.47275325295006906156677694749, 8.876481537351053215105899992619, 9.561173828168987741370687900859, 9.993240404916667514152826984381
