L(s) = 1 | + 3.42i·2-s + 69.6·3-s + 116.·4-s − 307. i·5-s + 238. i·6-s − 343i·7-s + 837. i·8-s + 2.66e3·9-s + 1.05e3·10-s − 4.37e3i·11-s + 8.09e3·12-s + (−3.94e3 + 6.86e3i)13-s + 1.17e3·14-s − 2.14e4i·15-s + 1.20e4·16-s + 3.77e4·17-s + ⋯ |
L(s) = 1 | + 0.303i·2-s + 1.48·3-s + 0.908·4-s − 1.10i·5-s + 0.451i·6-s − 0.377i·7-s + 0.578i·8-s + 1.21·9-s + 0.333·10-s − 0.991i·11-s + 1.35·12-s + (−0.497 + 0.867i)13-s + 0.114·14-s − 1.64i·15-s + 0.732·16-s + 1.86·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(4.04157 - 1.07775i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.04157 - 1.07775i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + 343iT \) |
| 13 | \( 1 + (3.94e3 - 6.86e3i)T \) |
good | 2 | \( 1 - 3.42iT - 128T^{2} \) |
| 3 | \( 1 - 69.6T + 2.18e3T^{2} \) |
| 5 | \( 1 + 307. iT - 7.81e4T^{2} \) |
| 11 | \( 1 + 4.37e3iT - 1.94e7T^{2} \) |
| 17 | \( 1 - 3.77e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.01e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 1.04e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 8.25e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.51e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 1.28e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 6.25e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 4.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.09e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 7.94e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.32e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.09e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.28e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 7.75e5iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 5.94e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 6.90e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.00e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 1.06e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 1.07e7iT - 8.07e13T^{2} \) |
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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
−12.68765407668210038841443147684, −11.69916140992651131359051412923, −10.15048242955914241800459247269, −8.969236717614221629937900501093, −8.140443446319824917430136565941, −7.22989831194697689241697910638, −5.58143497283447281223137624670, −3.87188846900590879954227870521, −2.55184334813339896498383615039, −1.18862760649827074587287255980,
1.84576191010529481200185279027, 2.76769888292958077283859109760, 3.61034874548469531715982565237, 6.00009888472450057745169286004, 7.50631196337417211678973072587, 7.931921973174919014317034430065, 9.959718254925337884969566008533, 10.10806229905807716670889381223, 11.83148194266452275021139944928, 12.64998844609153121759679435868