| L(s) = 1 | + (−0.956 − 1.04i)2-s + (−0.113 + 0.0225i)3-s + (−0.170 + 1.99i)4-s + (−0.831 − 0.555i)5-s + (0.131 + 0.0963i)6-s + (1.29 + 1.94i)7-s + (2.23 − 1.72i)8-s + (−2.75 + 1.14i)9-s + (0.216 + 1.39i)10-s + (0.268 − 1.35i)11-s + (−0.0255 − 0.229i)12-s + (−3.67 + 3.67i)13-s + (0.782 − 3.21i)14-s + (0.106 + 0.0441i)15-s + (−3.94 − 0.679i)16-s + (1.55 + 3.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.676 − 0.736i)2-s + (−0.0653 + 0.0129i)3-s + (−0.0852 + 0.996i)4-s + (−0.371 − 0.248i)5-s + (0.0537 + 0.0393i)6-s + (0.490 + 0.734i)7-s + (0.791 − 0.611i)8-s + (−0.919 + 0.380i)9-s + (0.0684 + 0.441i)10-s + (0.0810 − 0.407i)11-s + (−0.00737 − 0.0661i)12-s + (−1.01 + 1.01i)13-s + (0.209 − 0.858i)14-s + (0.0275 + 0.0113i)15-s + (−0.985 − 0.169i)16-s + (0.376 + 0.926i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.550040 + 0.311173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.550040 + 0.311173i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.956 + 1.04i)T \) |
| 5 | \( 1 + (0.831 + 0.555i)T \) |
| 17 | \( 1 + (-1.55 - 3.81i)T \) |
| good | 3 | \( 1 + (0.113 - 0.0225i)T + (2.77 - 1.14i)T^{2} \) |
| 7 | \( 1 + (-1.29 - 1.94i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.268 + 1.35i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (3.67 - 3.67i)T - 13iT^{2} \) |
| 19 | \( 1 + (1.89 - 4.56i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-9.11 - 1.81i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (3.61 - 5.41i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + (1.36 + 6.87i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (0.224 + 1.13i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (2.14 - 1.43i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-4.81 - 11.6i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-0.415 - 0.415i)T + 47iT^{2} \) |
| 53 | \( 1 + (6.23 + 2.58i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-9.07 + 3.75i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (0.482 + 0.721i)T + (-23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + 7.28T + 67T^{2} \) |
| 71 | \( 1 + (10.4 - 2.07i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.79 - 1.19i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (1.28 - 6.45i)T + (-72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (13.4 + 5.55i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (5.91 + 5.91i)T + 89iT^{2} \) |
| 97 | \( 1 + (2.26 - 3.39i)T + (-37.1 - 89.6i)T^{2} \) |
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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
−11.39642479600544517282080864953, −11.10547216724264767750814425664, −9.759323150247495203525739124537, −8.856612301722112823814345681998, −8.250096902993096580808791445197, −7.25820020246316554020508773501, −5.71485617293062243230431328288, −4.50943177793803156252091129087, −3.12102722327363489718813841824, −1.79575296142948624515264611005,
0.55619505482324789091240548651, 2.80665013554499457732294285354, 4.66635844271854580250888464924, 5.50167601895605306966334831938, 7.02310042708886841755546890835, 7.36382168737117305326529255726, 8.528651019642246483142021646832, 9.358773341092362168379203164817, 10.48454848968383034041101609917, 11.08192658921092422561134470768
