L(s) = 1 | + 3.23·2-s + (−3.88 − 3.45i)3-s + 2.45·4-s + (−8.12 + 7.68i)5-s + (−12.5 − 11.1i)6-s + (−17.9 − 4.72i)7-s − 17.9·8-s + (3.15 + 26.8i)9-s + (−26.2 + 24.8i)10-s − 0.605i·11-s + (−9.52 − 8.47i)12-s + 12.8·13-s + (−57.8 − 15.2i)14-s + (58.0 − 1.78i)15-s − 77.6·16-s − 117. i·17-s + ⋯ |
L(s) = 1 | + 1.14·2-s + (−0.747 − 0.664i)3-s + 0.306·4-s + (−0.726 + 0.687i)5-s + (−0.854 − 0.759i)6-s + (−0.966 − 0.255i)7-s − 0.792·8-s + (0.116 + 0.993i)9-s + (−0.830 + 0.785i)10-s − 0.0165i·11-s + (−0.229 − 0.203i)12-s + 0.273·13-s + (−1.10 − 0.291i)14-s + (0.999 − 0.0307i)15-s − 1.21·16-s − 1.68i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.00452569 + 0.0396441i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00452569 + 0.0396441i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (3.88 + 3.45i)T \) |
| 5 | \( 1 + (8.12 - 7.68i)T \) |
| 7 | \( 1 + (17.9 + 4.72i)T \) |
good | 2 | \( 1 - 3.23T + 8T^{2} \) |
| 11 | \( 1 + 0.605iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 12.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 98.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 77.5iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 131. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 260. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 58.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 519. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 104. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 550.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 498.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 172. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 622. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 151. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 242.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 94.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 779. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−12.62821338277481270948486675344, −12.01445554347097080279037151721, −11.05568704473019867726583899645, −9.713521806255375698728848602720, −7.81858887846808129791385116867, −6.68451459574411064191533894300, −5.82547834281223379715086509868, −4.30331256216656976884987218393, −2.97837963286111582570983185897, −0.01625805401571644118100025150,
3.51512186570352414269651317308, 4.36417468378424056367041202633, 5.58174125610357738728920791219, 6.53008738612325196760596886427, 8.568776281929667269932947368832, 9.608700391959825759158157032991, 10.98458923860592489254161149833, 12.13636206284515491220764132908, 12.61579867722300795311694605232, 13.61275400153369805816243501191