| L(s) = 1 | + (−0.122 − 0.475i)2-s + (0.685 − 0.130i)3-s + (1.54 − 0.847i)4-s + (−0.146 − 0.310i)6-s + (−0.173 + 0.238i)7-s + (−1.26 − 1.34i)8-s + (−2.33 + 0.924i)9-s + (2.89 − 0.742i)11-s + (0.946 − 0.782i)12-s + (−0.880 − 2.22i)13-s + (0.134 + 0.0533i)14-s + (1.39 − 2.20i)16-s + (3.17 − 5.77i)17-s + (0.725 + 0.998i)18-s + (0.342 − 1.79i)19-s + ⋯ |
| L(s) = 1 | + (−0.0863 − 0.336i)2-s + (0.396 − 0.0755i)3-s + (0.770 − 0.423i)4-s + (−0.0596 − 0.126i)6-s + (−0.0655 + 0.0902i)7-s + (−0.446 − 0.475i)8-s + (−0.778 + 0.308i)9-s + (0.872 − 0.223i)11-s + (0.273 − 0.225i)12-s + (−0.244 − 0.616i)13-s + (0.0360 + 0.0142i)14-s + (0.349 − 0.550i)16-s + (0.769 − 1.39i)17-s + (0.171 + 0.235i)18-s + (0.0785 − 0.411i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48432 - 1.12357i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48432 - 1.12357i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (0.122 + 0.475i)T + (-1.75 + 0.963i)T^{2} \) |
| 3 | \( 1 + (-0.685 + 0.130i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (0.173 - 0.238i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-2.89 + 0.742i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (0.880 + 2.22i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (-3.17 + 5.77i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (-0.342 + 1.79i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (-8.69 - 0.546i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (9.54 + 1.20i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (-7.45 - 4.10i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (-0.0198 - 0.0126i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-0.144 - 2.29i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (2.00 - 0.651i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (1.59 - 1.69i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-0.713 - 0.335i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (0.619 + 0.748i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (0.485 - 7.71i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (-0.819 - 6.48i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (6.47 + 6.07i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-0.364 - 0.301i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (-0.667 - 3.49i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (13.3 + 2.54i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (-6.85 + 8.28i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (1.73 - 13.7i)T + (-93.9 - 24.1i)T^{2} \) |
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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
−10.50747988162869086921847696466, −9.487990280876456184675829186023, −8.939442117035365958628303527184, −7.67445786385372890149974154836, −6.97065340543710178000134036805, −5.86657878798037873029670917893, −5.02755754643214968930254997727, −3.24163077170966325104674627917, −2.67574039143062124162434248545, −1.07822695117551215011677838882,
1.79210845979704557692228531160, 3.12835999301203765519509180824, 3.95992378332159460328169298360, 5.56845864357318460479296780718, 6.42398404133101977100709881834, 7.22467462282389458654556461324, 8.187040991208635153985510519446, 8.886122168691870518924331352111, 9.759940260656797436146933762378, 10.93223363925460024292609689200
