L(s) = 1 | + (0.539 + 0.933i)2-s + (−1.46 + 0.918i)3-s + (0.418 − 0.725i)4-s + (−1.64 − 0.876i)6-s + (−0.929 + 2.47i)7-s + 3.05·8-s + (1.31 − 2.69i)9-s + (3.84 + 2.21i)11-s + (0.0513 + 1.44i)12-s − 0.955·13-s + (−2.81 + 0.467i)14-s + (0.812 + 1.40i)16-s + (0.439 + 0.253i)17-s + (3.22 − 0.228i)18-s + (−4.41 + 2.54i)19-s + ⋯ |
L(s) = 1 | + (0.381 + 0.660i)2-s + (−0.847 + 0.530i)3-s + (0.209 − 0.362i)4-s + (−0.673 − 0.357i)6-s + (−0.351 + 0.936i)7-s + 1.08·8-s + (0.437 − 0.899i)9-s + (1.15 + 0.669i)11-s + (0.0148 + 0.418i)12-s − 0.265·13-s + (−0.752 + 0.125i)14-s + (0.203 + 0.351i)16-s + (0.106 + 0.0615i)17-s + (0.760 − 0.0539i)18-s + (−1.01 + 0.584i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.884253 + 1.15854i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.884253 + 1.15854i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.46 - 0.918i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.929 - 2.47i)T \) |
good | 2 | \( 1 + (-0.539 - 0.933i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-3.84 - 2.21i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 0.955T + 13T^{2} \) |
| 17 | \( 1 + (-0.439 - 0.253i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.41 - 2.54i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.14 - 3.72i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.89iT - 29T^{2} \) |
| 31 | \( 1 + (-5.10 - 2.94i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.51 - 3.76i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.65T + 41T^{2} \) |
| 43 | \( 1 + 0.492iT - 43T^{2} \) |
| 47 | \( 1 + (-5.76 + 3.32i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.56 + 7.90i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.81 + 10.0i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.399 + 0.230i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.20 + 1.85i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.90iT - 71T^{2} \) |
| 73 | \( 1 + (-3.15 + 5.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.38 - 12.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.7iT - 83T^{2} \) |
| 89 | \( 1 + (-3.57 - 6.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.91T + 97T^{2} \) |
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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
−11.11044540834016429935569657234, −10.20194846287377811402262548219, −9.525627731441502055770003302908, −8.508523610994249103584651260463, −6.88038322237146714865031614423, −6.57753015573444007342990639899, −5.51021936201585234619698073027, −4.86188116421719040707242831505, −3.66793510738813093285796766496, −1.68626428707863088681843289657,
0.913089128559564419405041608863, 2.44104008240702619781872913263, 3.88849723630060169985630274281, 4.60626016213918595936476287579, 6.15324105884675616990229792800, 6.85763848493406115195606954361, 7.65523212419922826482308049898, 8.804652856364655295083344554684, 10.24420653605710261412377978049, 10.74840132782737004077684930656