L(s) = 1 | − 1.84i·2-s + (−1.59 − 0.682i)3-s − 1.42·4-s − 2.40·5-s + (−1.26 + 2.94i)6-s + (−1.26 + 2.32i)7-s − 1.07i·8-s + (2.06 + 2.17i)9-s + 4.43i·10-s + 3.68i·11-s + (2.26 + 0.969i)12-s − 2.24i·13-s + (4.29 + 2.34i)14-s + (3.82 + 1.63i)15-s − 4.82·16-s + 17-s + ⋯ |
L(s) = 1 | − 1.30i·2-s + (−0.919 − 0.394i)3-s − 0.710·4-s − 1.07·5-s + (−0.515 + 1.20i)6-s + (−0.478 + 0.878i)7-s − 0.379i·8-s + (0.689 + 0.724i)9-s + 1.40i·10-s + 1.10i·11-s + (0.652 + 0.279i)12-s − 0.622i·13-s + (1.14 + 0.625i)14-s + (0.986 + 0.423i)15-s − 1.20·16-s + 0.242·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.286062 + 0.0991116i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.286062 + 0.0991116i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.59 + 0.682i)T \) |
| 7 | \( 1 + (1.26 - 2.32i)T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + 1.84iT - 2T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 11 | \( 1 - 3.68iT - 11T^{2} \) |
| 13 | \( 1 + 2.24iT - 13T^{2} \) |
| 19 | \( 1 - 4.04iT - 19T^{2} \) |
| 23 | \( 1 - 0.471iT - 23T^{2} \) |
| 29 | \( 1 - 1.76iT - 29T^{2} \) |
| 31 | \( 1 - 5.78iT - 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 3.80T + 43T^{2} \) |
| 47 | \( 1 - 9.80T + 47T^{2} \) |
| 53 | \( 1 - 13.4iT - 53T^{2} \) |
| 59 | \( 1 + 8.69T + 59T^{2} \) |
| 61 | \( 1 - 4.34iT - 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 4.79iT - 71T^{2} \) |
| 73 | \( 1 + 4.36iT - 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 7.25T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 0.875iT - 97T^{2} \) |
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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
−11.88128386958201722115244763104, −10.68329122583483213196131750900, −10.22586107883040811649906864169, −9.040065537096812799031516788735, −7.71154189526804294648376798791, −6.83748472167763283223077836342, −5.54111391638344406746154233181, −4.32976118418626443041870373750, −3.14502768528119614697195196180, −1.66981423291780553826170297527,
0.23199994682243059839310006498, 3.61375800715521234446459826262, 4.55762312146611598592572737811, 5.68759196152239413776341079545, 6.67279802055152919531618336679, 7.26315956264416894158321081580, 8.286746855103560262021794267207, 9.330548547925647568772563619770, 10.62820523449257487156361322462, 11.32752256589622939139541184896