L(s) = 1 | + 4·2-s + (1.66 − 15.4i)3-s + 16·4-s + 65.2i·5-s + (6.65 − 61.9i)6-s − 43.9·7-s + 64·8-s + (−237. − 51.5i)9-s + 261. i·10-s + 184.·11-s + (26.6 − 247. i)12-s + 483. i·13-s − 175.·14-s + (1.01e3 + 108. i)15-s + 256·16-s − 1.66e3i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.106 − 0.994i)3-s + 0.5·4-s + 1.16i·5-s + (0.0754 − 0.703i)6-s − 0.339·7-s + 0.353·8-s + (−0.977 − 0.212i)9-s + 0.825i·10-s + 0.459·11-s + (0.0533 − 0.497i)12-s + 0.792i·13-s − 0.239·14-s + (1.16 + 0.124i)15-s + 0.250·16-s − 1.39i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.385898113\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.385898113\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + (-1.66 + 15.4i)T \) |
| 59 | \( 1 + (-2.62e4 - 4.88e3i)T \) |
good | 5 | \( 1 - 65.2iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 43.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 184.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 483. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.66e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 464.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.98e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.84e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 9.47e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 4.96e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.88e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 3.62e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 7.49e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.88e3iT - 4.18e8T^{2} \) |
| 61 | \( 1 + 3.76e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.94e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.55e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 7.20e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.62e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.57e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.00e4iT - 8.58e9T^{2} \) |
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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
−11.21980963578921853076851415815, −10.04993835268507844982168113214, −8.902471567490237836282798353088, −7.58318980857188004291834234283, −6.77963088078221368265165994369, −6.38908152089629050932543118434, −5.00600225136452758251071884749, −3.41150415804548362116263745321, −2.69820776613331903277374145425, −1.40904233494345517882657741787,
0.45014815102553518399889687054, 2.18253091994187503304613584915, 3.75384793588386887832190599946, 4.25618139390367265563918459885, 5.51166737218198134263742961310, 6.05098829983028168452363503322, 7.83458169839817704814363312330, 8.624353728648722112378628035572, 9.647314365665696492853704735970, 10.37946197017653687546964031063