| L(s) = 1 | + (−0.654 − 0.755i)2-s + (0.124 + 0.0797i)3-s + (−0.142 + 0.989i)4-s + (1.52 − 3.32i)5-s + (−0.0209 − 0.145i)6-s + (0.959 + 0.281i)7-s + (0.841 − 0.540i)8-s + (−1.23 − 2.70i)9-s + (−3.51 + 1.03i)10-s + (−3.04 + 3.51i)11-s + (−0.0965 + 0.111i)12-s + (4.26 − 1.25i)13-s + (−0.415 − 0.909i)14-s + (0.454 − 0.291i)15-s + (−0.959 − 0.281i)16-s + (−0.656 − 4.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (0.0716 + 0.0460i)3-s + (−0.0711 + 0.494i)4-s + (0.680 − 1.48i)5-s + (−0.00856 − 0.0595i)6-s + (0.362 + 0.106i)7-s + (0.297 − 0.191i)8-s + (−0.412 − 0.903i)9-s + (−1.11 + 0.326i)10-s + (−0.917 + 1.05i)11-s + (−0.0278 + 0.0321i)12-s + (1.18 − 0.347i)13-s + (−0.111 − 0.243i)14-s + (0.117 − 0.0753i)15-s + (−0.239 − 0.0704i)16-s + (−0.159 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.707159 - 0.877826i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.707159 - 0.877826i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (0.962 + 4.69i)T \) |
| good | 3 | \( 1 + (-0.124 - 0.0797i)T + (1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (-1.52 + 3.32i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (3.04 - 3.51i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-4.26 + 1.25i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.656 + 4.56i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.587 - 4.08i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.882 + 6.13i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.96 + 3.83i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-4.28 - 9.39i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (3.95 - 8.66i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (5.20 + 3.34i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 + (-12.7 - 3.72i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-0.348 + 0.102i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-3.22 + 2.07i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-6.52 - 7.52i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-6.69 - 7.72i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.945 - 6.57i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-1.40 + 0.411i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.634 + 1.39i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-7.67 - 4.92i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (6.69 - 14.6i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.55764325528856301768861056006, −10.05177104073143205125389142812, −9.674397813882947019934923587283, −8.490617287408920154234020658800, −8.133170131718579933840258355818, −6.38093628151581732663105035677, −5.22858339561504931732583559280, −4.22182795156920128004740856637, −2.47450700626964239950103310732, −0.993664721959160161825676067180,
2.06070042952033253737044152558, 3.38789835924799791130986766715, 5.33299976796361312891379767437, 6.12861987345664290502006124647, 7.06629437643775332067388520348, 8.119004038561099759621496183707, 8.854378631441990836660837264413, 10.29662496668244449825027246611, 10.84659312039034664085425570472, 11.26401954612116985638449248272
