| L(s) = 1 | + (0.810 − 1.15i)2-s + (−1.01 + 0.542i)3-s + (−0.685 − 1.87i)4-s + (0.634 + 0.773i)5-s + (−0.194 + 1.61i)6-s + (−1.73 + 1.15i)7-s + (−2.73 − 0.728i)8-s + (−0.930 + 1.39i)9-s + (1.41 − 0.108i)10-s + (−0.0953 + 0.0289i)11-s + (1.71 + 1.53i)12-s + (5.37 + 4.40i)13-s + (−0.0631 + 2.94i)14-s + (−1.06 − 0.440i)15-s + (−3.06 + 2.57i)16-s + (2.51 − 1.04i)17-s + ⋯ |
| L(s) = 1 | + (0.573 − 0.819i)2-s + (−0.586 + 0.313i)3-s + (−0.342 − 0.939i)4-s + (0.283 + 0.345i)5-s + (−0.0792 + 0.659i)6-s + (−0.655 + 0.438i)7-s + (−0.966 − 0.257i)8-s + (−0.310 + 0.464i)9-s + (0.445 − 0.0342i)10-s + (−0.0287 + 0.00871i)11-s + (0.495 + 0.443i)12-s + (1.48 + 1.22i)13-s + (−0.0168 + 0.788i)14-s + (−0.274 − 0.113i)15-s + (−0.765 + 0.644i)16-s + (0.609 − 0.252i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24949 + 0.377234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24949 + 0.377234i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.810 + 1.15i)T \) |
| 5 | \( 1 + (-0.634 - 0.773i)T \) |
| good | 3 | \( 1 + (1.01 - 0.542i)T + (1.66 - 2.49i)T^{2} \) |
| 7 | \( 1 + (1.73 - 1.15i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (0.0953 - 0.0289i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-5.37 - 4.40i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-2.51 + 1.04i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.383 - 3.89i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (3.00 - 0.597i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (0.430 - 1.41i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-5.84 - 5.84i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.21 + 0.217i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.994 - 5.00i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (1.07 + 0.575i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (2.33 + 5.64i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-0.222 - 0.735i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (1.01 - 0.834i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (0.862 + 1.61i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (5.00 + 9.36i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-5.05 - 7.56i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-5.49 - 3.67i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.09 + 5.06i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (15.0 - 1.48i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-16.6 - 3.32i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (1.75 + 1.75i)T + 97iT^{2} \) |
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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
−10.70480417095886657836412210777, −10.08217784549266112877502887440, −9.243536216453658149083521763566, −8.276825171188891440778533131956, −6.54855652653108119253120387214, −6.02937026288434087591531708453, −5.15196300984626270074787929381, −3.98414766739014511395968669312, −3.00649930559450576097158803600, −1.63896042412214901026510654484,
0.65944206455715671137956963354, 3.06336068583343485566466627311, 3.97574173222806785981851880152, 5.32919027667094336240936447115, 6.05305873500612835916192502446, 6.57053569823920268681155373012, 7.74852436086061859810682514203, 8.536762938447101420382479227955, 9.482755452473597566984167270328, 10.55656981000154771360639880357
