| L(s) = 1 | + (1.37 + 0.335i)2-s + (−1.32 + 0.669i)3-s + (1.77 + 0.921i)4-s + (0.121 + 0.274i)5-s + (−2.05 + 0.473i)6-s + (−2.20 − 1.32i)7-s + (2.12 + 1.86i)8-s + (−0.466 + 0.629i)9-s + (0.0750 + 0.417i)10-s + (4.27 + 3.33i)11-s + (−2.97 − 0.0372i)12-s + (−3.22 + 3.07i)13-s + (−2.58 − 2.55i)14-s + (−0.345 − 0.283i)15-s + (2.30 + 3.27i)16-s + (3.34 + 4.07i)17-s + ⋯ |
| L(s) = 1 | + (0.971 + 0.237i)2-s + (−0.767 + 0.386i)3-s + (0.887 + 0.460i)4-s + (0.0543 + 0.122i)5-s + (−0.837 + 0.193i)6-s + (−0.833 − 0.499i)7-s + (0.752 + 0.658i)8-s + (−0.155 + 0.209i)9-s + (0.0237 + 0.132i)10-s + (1.28 + 1.00i)11-s + (−0.859 − 0.0107i)12-s + (−0.894 + 0.851i)13-s + (−0.691 − 0.683i)14-s + (−0.0891 − 0.0731i)15-s + (0.575 + 0.817i)16-s + (0.811 + 0.988i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15576 + 1.39805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15576 + 1.39805i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.37 - 0.335i)T \) |
| good | 3 | \( 1 + (1.32 - 0.669i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-0.121 - 0.274i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (2.20 + 1.32i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-4.27 - 3.33i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (3.22 - 3.07i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-3.34 - 4.07i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-0.260 - 1.50i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (5.31 + 2.51i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-1.29 - 0.733i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (5.91 - 1.17i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-2.02 - 1.27i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-6.22 - 6.86i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (3.60 + 10.9i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (2.68 + 1.43i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (-10.5 + 5.97i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (-6.62 + 6.95i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.468 - 6.35i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (-3.05 - 2.63i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-3.63 - 0.538i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-9.85 + 5.90i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (-3.36 - 1.02i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-3.59 + 2.27i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (14.9 - 7.08i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (9.35 + 6.25i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.34099557143954138688752166413, −10.27732231185738666710428060705, −9.798039369040153228408140799613, −8.271942719448043024768547089379, −6.99430125352540206816772198047, −6.53121756777721499874011211988, −5.52265342198633567353989171657, −4.41491599292087370805115687827, −3.76335741908011940370332107742, −2.09542239401419574863440455655,
0.886031931299314489161594551649, 2.81158775159282103639129502995, 3.70522698995273906410822499337, 5.29458355933154894632350141564, 5.83019063608357510199883834787, 6.62321568693465149956603341412, 7.54813162088454441016918486345, 9.148791185204736604129599526720, 9.829982510927971891119631764663, 11.11235267975112547462763320028
