L(s) = 1 | + 4·2-s + (−11.5 + 10.4i)3-s + 16·4-s + 67.9i·5-s + (−46.0 + 41.9i)6-s − 61.6·7-s + 64·8-s + (22.5 − 241. i)9-s + 271. i·10-s + 640.·11-s + (−184. + 167. i)12-s − 578. i·13-s − 246.·14-s + (−713. − 783. i)15-s + 256·16-s − 1.37e3i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.739 + 0.673i)3-s + 0.5·4-s + 1.21i·5-s + (−0.522 + 0.476i)6-s − 0.475·7-s + 0.353·8-s + (0.0928 − 0.995i)9-s + 0.859i·10-s + 1.59·11-s + (−0.369 + 0.336i)12-s − 0.948i·13-s − 0.336·14-s + (−0.819 − 0.899i)15-s + 0.250·16-s − 1.15i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.852030142\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.852030142\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + (11.5 - 10.4i)T \) |
| 59 | \( 1 + (2.67e4 + 479. i)T \) |
good | 5 | \( 1 - 67.9iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 61.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 640.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 578. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.37e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.52e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 625.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.93e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 491. iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.35e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 780. iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 4.53e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.83e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.72e4iT - 4.18e8T^{2} \) |
| 61 | \( 1 - 854. iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.85e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.29e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 5.07e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 9.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.49e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.22e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 828. iT - 8.58e9T^{2} \) |
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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
−10.92079442243139400301727425153, −9.958904474977909341763060959881, −9.330410816132863719462189604812, −7.48210875765575559164023991114, −6.62768740995401421967327501098, −5.95604144123867885097645225522, −4.81864152385804198140459607682, −3.56990178629592389261791003252, −2.95128844902441721243656247595, −0.852388977364535115054173839409,
0.976427951987151853975549330436, 1.71249205936259147834240778328, 3.66070515936897012940371466651, 4.68360996963664062594395695688, 5.65756355789908286478937776918, 6.52778779013843312819495239784, 7.36447059395694396659701114131, 8.727980398002218570876020955073, 9.508038664014965540412066247242, 10.91604578656542590491137570298