| L(s) = 1 | + (0.125 − 0.992i)2-s + (2.11 − 1.74i)3-s + (−0.968 − 0.248i)4-s + (2.06 − 0.850i)5-s + (−1.46 − 2.31i)6-s + (−3.77 + 1.22i)7-s + (−0.368 + 0.929i)8-s + (0.844 − 4.42i)9-s + (−0.584 − 2.15i)10-s + (1.18 + 0.149i)11-s + (−2.47 + 1.16i)12-s + (1.33 + 0.255i)13-s + (0.744 + 3.90i)14-s + (2.87 − 5.40i)15-s + (0.876 + 0.481i)16-s + (1.09 + 4.27i)17-s + ⋯ |
| L(s) = 1 | + (0.0886 − 0.701i)2-s + (1.21 − 1.00i)3-s + (−0.484 − 0.124i)4-s + (0.924 − 0.380i)5-s + (−0.599 − 0.944i)6-s + (−1.42 + 0.463i)7-s + (−0.130 + 0.328i)8-s + (0.281 − 1.47i)9-s + (−0.184 − 0.682i)10-s + (0.357 + 0.0452i)11-s + (−0.715 + 0.336i)12-s + (0.370 + 0.0707i)13-s + (0.198 + 1.04i)14-s + (0.743 − 1.39i)15-s + (0.219 + 0.120i)16-s + (0.266 + 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12234 - 1.42495i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12234 - 1.42495i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.125 + 0.992i)T \) |
| 5 | \( 1 + (-2.06 + 0.850i)T \) |
| good | 3 | \( 1 + (-2.11 + 1.74i)T + (0.562 - 2.94i)T^{2} \) |
| 7 | \( 1 + (3.77 - 1.22i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (-1.18 - 0.149i)T + (10.6 + 2.73i)T^{2} \) |
| 13 | \( 1 + (-1.33 - 0.255i)T + (12.0 + 4.78i)T^{2} \) |
| 17 | \( 1 + (-1.09 - 4.27i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (4.73 - 5.72i)T + (-3.56 - 18.6i)T^{2} \) |
| 23 | \( 1 + (-4.40 + 4.69i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.138 + 2.19i)T + (-28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (6.46 - 1.65i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (-4.30 + 7.83i)T + (-19.8 - 31.2i)T^{2} \) |
| 41 | \( 1 + (4.94 - 4.64i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-1.20 - 1.66i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-1.69 - 4.27i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (6.29 + 3.99i)T + (22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (-3.61 - 7.68i)T + (-37.6 + 45.4i)T^{2} \) |
| 61 | \( 1 + (5.15 + 4.84i)T + (3.83 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-2.91 - 0.183i)T + (66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (2.99 - 1.18i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (-13.1 - 6.16i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (0.244 + 0.295i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (13.0 + 10.7i)T + (15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (3.73 - 7.94i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (-8.56 + 0.539i)T + (96.2 - 12.1i)T^{2} \) |
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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
−12.50641057690610069825937857197, −10.66870168053158217585937022010, −9.644636554711664005707039185695, −8.967385088713186494301980544686, −8.218060144858426400865704900885, −6.63186862689543998673286272864, −5.91193487566191176398589642254, −3.83593612286813611849232208051, −2.69273751416059115497836293034, −1.62239973024093733982475526482,
2.83966733984302258375354419226, 3.67957236843231879121684891855, 5.06681824788783948402202040977, 6.45917487832302583623942602581, 7.25075785326224182729141636670, 8.831355829778122825853226956429, 9.350688458221045621342382233233, 9.964854599909474561513914898544, 10.98958136271407478147274777686, 12.94260402659010552892283120525
