| L(s) = 1 | + (−0.726 + 0.686i)2-s + (0.683 + 3.23i)3-s + (0.0567 − 0.998i)4-s + (−0.0475 − 2.51i)5-s + (−2.72 − 1.88i)6-s + (1.57 + 2.01i)7-s + (0.644 + 0.764i)8-s + (−7.27 + 3.21i)9-s + (1.76 + 1.79i)10-s + (−1.95 + 1.14i)11-s + (3.27 − 0.499i)12-s + (−6.48 + 2.58i)13-s + (−2.52 − 0.385i)14-s + (8.10 − 1.87i)15-s + (−0.993 − 0.113i)16-s + (4.71 + 3.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.513 + 0.485i)2-s + (0.394 + 1.86i)3-s + (0.0283 − 0.499i)4-s + (−0.0212 − 1.12i)5-s + (−1.11 − 0.769i)6-s + (0.594 + 0.762i)7-s + (0.227 + 0.270i)8-s + (−2.42 + 1.07i)9-s + (0.556 + 0.567i)10-s + (−0.587 + 0.344i)11-s + (0.944 − 0.144i)12-s + (−1.79 + 0.715i)13-s + (−0.675 − 0.103i)14-s + (2.09 − 0.483i)15-s + (−0.248 − 0.0283i)16-s + (1.14 + 0.728i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.117696 + 0.961261i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.117696 + 0.961261i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.726 - 0.686i)T \) |
| 167 | \( 1 + (-1.61 + 12.8i)T \) |
| good | 3 | \( 1 + (-0.683 - 3.23i)T + (-2.74 + 1.21i)T^{2} \) |
| 5 | \( 1 + (0.0475 + 2.51i)T + (-4.99 + 0.189i)T^{2} \) |
| 7 | \( 1 + (-1.57 - 2.01i)T + (-1.70 + 6.78i)T^{2} \) |
| 11 | \( 1 + (1.95 - 1.14i)T + (5.37 - 9.59i)T^{2} \) |
| 13 | \( 1 + (6.48 - 2.58i)T + (9.44 - 8.92i)T^{2} \) |
| 17 | \( 1 + (-4.71 - 3.00i)T + (7.16 + 15.4i)T^{2} \) |
| 19 | \( 1 + (1.84 - 3.60i)T + (-11.1 - 15.4i)T^{2} \) |
| 23 | \( 1 + (-6.11 + 1.90i)T + (18.9 - 13.0i)T^{2} \) |
| 29 | \( 1 + (2.83 + 0.886i)T + (23.8 + 16.5i)T^{2} \) |
| 31 | \( 1 + (0.542 + 4.07i)T + (-29.9 + 8.11i)T^{2} \) |
| 37 | \( 1 + (-6.93 - 3.06i)T + (24.8 + 27.3i)T^{2} \) |
| 41 | \( 1 + (-3.17 + 0.862i)T + (35.3 - 20.7i)T^{2} \) |
| 43 | \( 1 + (2.24 - 0.170i)T + (42.5 - 6.48i)T^{2} \) |
| 47 | \( 1 + (-9.41 - 7.06i)T + (13.1 + 45.1i)T^{2} \) |
| 53 | \( 1 + (-3.42 + 1.83i)T + (29.3 - 44.1i)T^{2} \) |
| 59 | \( 1 + (-2.53 + 1.62i)T + (24.8 - 53.4i)T^{2} \) |
| 61 | \( 1 + (0.290 - 0.871i)T + (-48.8 - 36.5i)T^{2} \) |
| 67 | \( 1 + (-0.145 + 7.70i)T + (-66.9 - 2.53i)T^{2} \) |
| 71 | \( 1 + (-8.39 - 7.34i)T + (9.37 + 70.3i)T^{2} \) |
| 73 | \( 1 + (10.8 - 1.23i)T + (71.1 - 16.4i)T^{2} \) |
| 79 | \( 1 + (-10.5 - 3.74i)T + (61.3 + 49.7i)T^{2} \) |
| 83 | \( 1 + (-0.129 - 0.122i)T + (4.70 + 82.8i)T^{2} \) |
| 89 | \( 1 + (4.00 + 0.925i)T + (79.9 + 39.0i)T^{2} \) |
| 97 | \( 1 + (0.0717 - 0.538i)T + (-93.6 - 25.4i)T^{2} \) |
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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
−11.81219294406015274817134452304, −10.70394233904279951485171141592, −9.781253071312958745097173202030, −9.311285824001888704237874816353, −8.460355498073946553654506970300, −7.75170928814285074720956570515, −5.63419879398033619281149082918, −5.01989259000747921543548596410, −4.28200351455711434205113593468, −2.43124678868022497080853072641,
0.76433168766087696457038781384, 2.47363108958515855469423608181, 3.04825018086542026393477338866, 5.35449820088457539773582238673, 6.96824958419846274374362390760, 7.37693110404926727869280614670, 7.87997071577110781480874019446, 9.166682452790109260314183168077, 10.44564494651848378140674576490, 11.18372150358776302645486052478
