L(s) = 1 | + 2.69i·2-s − 3-s − 5.24·4-s − 1.04i·5-s − 2.69i·6-s − 0.554i·7-s − 8.74i·8-s + 9-s + 2.82·10-s − 2.91i·11-s + 5.24·12-s + 1.49·14-s + 1.04i·15-s + 13.0·16-s + 4.85·17-s + 2.69i·18-s + ⋯ |
L(s) = 1 | + 1.90i·2-s − 0.577·3-s − 2.62·4-s − 0.469i·5-s − 1.09i·6-s − 0.209i·7-s − 3.09i·8-s + 0.333·9-s + 0.892·10-s − 0.877i·11-s + 1.51·12-s + 0.399·14-s + 0.270i·15-s + 3.25·16-s + 1.17·17-s + 0.634i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.762762 + 0.325622i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.762762 + 0.325622i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.69iT - 2T^{2} \) |
| 5 | \( 1 + 1.04iT - 5T^{2} \) |
| 7 | \( 1 + 0.554iT - 7T^{2} \) |
| 11 | \( 1 + 2.91iT - 11T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 0.753iT - 19T^{2} \) |
| 23 | \( 1 + 5.76T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 + 9.51iT - 31T^{2} \) |
| 37 | \( 1 + 5.75iT - 37T^{2} \) |
| 41 | \( 1 - 4.91iT - 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 0.753iT - 47T^{2} \) |
| 53 | \( 1 + 7.58T + 53T^{2} \) |
| 59 | \( 1 + 4.09iT - 59T^{2} \) |
| 61 | \( 1 + 3.42T + 61T^{2} \) |
| 67 | \( 1 + 1.87iT - 67T^{2} \) |
| 71 | \( 1 + 10.5iT - 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 1.33T + 79T^{2} \) |
| 83 | \( 1 - 2.64iT - 83T^{2} \) |
| 89 | \( 1 + 9.92iT - 89T^{2} \) |
| 97 | \( 1 - 17.0iT - 97T^{2} \) |
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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
−10.87704805588224926532475307568, −9.745704545108987706096917789871, −9.023076941521660372795479791779, −7.968082557174193774427173795837, −7.47890302062870741140047292776, −6.17014533970443346282452374444, −5.78404864299300655338576803600, −4.76981986409579124471295970560, −3.78191431444966897064624159984, −0.61856969494096997661188266923,
1.35479947325132124480639002243, 2.62928793339349890588837605010, 3.74934270031281995409354816353, 4.77739702292958854066591356164, 5.76165902909497038600322071581, 7.27458693931038230018875232387, 8.476858772976323675348507410648, 9.538461493695778912183071773518, 10.22733530994332378678351123597, 10.73873219059207446410156089213