| L(s) = 1 | + (−1.93 + 1.24i)2-s + (−0.142 + 0.989i)3-s + (1.35 − 2.97i)4-s + (0.959 − 0.281i)5-s + (−0.953 − 2.08i)6-s + (−1.49 − 1.72i)7-s + (0.416 + 2.89i)8-s + (−0.959 − 0.281i)9-s + (−1.50 + 1.73i)10-s + (4.62 + 2.97i)11-s + (2.75 + 1.76i)12-s + (−0.384 + 0.443i)13-s + (5.03 + 1.47i)14-s + (0.142 + 0.989i)15-s + (−0.110 − 0.127i)16-s + (1.85 + 4.07i)17-s + ⋯ |
| L(s) = 1 | + (−1.36 + 0.877i)2-s + (−0.0821 + 0.571i)3-s + (0.679 − 1.48i)4-s + (0.429 − 0.125i)5-s + (−0.389 − 0.852i)6-s + (−0.565 − 0.653i)7-s + (0.147 + 1.02i)8-s + (−0.319 − 0.0939i)9-s + (−0.475 + 0.548i)10-s + (1.39 + 0.896i)11-s + (0.794 + 0.510i)12-s + (−0.106 + 0.122i)13-s + (1.34 + 0.395i)14-s + (0.0367 + 0.255i)15-s + (−0.0276 − 0.0318i)16-s + (0.450 + 0.987i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.444157 + 0.532443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.444157 + 0.532443i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (-1.05 - 4.67i)T \) |
| good | 2 | \( 1 + (1.93 - 1.24i)T + (0.830 - 1.81i)T^{2} \) |
| 7 | \( 1 + (1.49 + 1.72i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-4.62 - 2.97i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.384 - 0.443i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.85 - 4.07i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.52 + 5.52i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.12 - 2.46i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.733 - 5.10i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-4.09 - 1.20i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (6.89 - 2.02i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.18 - 8.27i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 2.69T + 47T^{2} \) |
| 53 | \( 1 + (6.98 + 8.06i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.41 + 2.78i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.25 - 8.70i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-10.2 + 6.58i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.96 + 3.83i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.44 + 3.16i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.740 + 0.854i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-15.8 - 4.66i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (2.53 - 17.6i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (0.126 - 0.0372i)T + (81.6 - 52.4i)T^{2} \) |
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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
−11.38419561953532643458921553627, −10.35666173789605540716489451084, −9.578166840598950133750206747948, −9.274843889707373680386998872941, −8.113144475759978220218712056758, −6.90502981486034923062224002915, −6.48751903025486365076856105874, −5.06573851454675248170780621341, −3.64038234667538826644544763855, −1.32028229188747585272966478138,
0.914951855226830235141363695844, 2.33944479197629778783492513621, 3.43860800847949842904201432115, 5.66822712496240872802650874484, 6.59619285858319301004495869593, 7.78380483266783116045740373348, 8.741115261760575537810744743419, 9.418470321156578198204925736539, 10.15127191099967812582839389167, 11.29036656636815784352245758732
