L(s) = 1 | + (1 + i)2-s + (2.74 − 1.21i)3-s + 2i·4-s + (3.32 − 3.73i)5-s + (3.95 + 1.52i)6-s + (2.29 + 6.61i)7-s + (−2 + 2i)8-s + (6.03 − 6.67i)9-s + (7.05 − 0.417i)10-s − 10.5i·11-s + (2.43 + 5.48i)12-s + (−14.9 + 14.9i)13-s + (−4.31 + 8.90i)14-s + (4.55 − 14.2i)15-s − 4·16-s + (15.4 − 15.4i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (0.913 − 0.405i)3-s + 0.5i·4-s + (0.664 − 0.747i)5-s + (0.659 + 0.254i)6-s + (0.327 + 0.944i)7-s + (−0.250 + 0.250i)8-s + (0.670 − 0.741i)9-s + (0.705 − 0.0417i)10-s − 0.961i·11-s + (0.202 + 0.456i)12-s + (−1.15 + 1.15i)13-s + (−0.308 + 0.636i)14-s + (0.303 − 0.952i)15-s − 0.250·16-s + (0.911 − 0.911i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.85740 + 0.369815i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.85740 + 0.369815i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 + (-2.74 + 1.21i)T \) |
| 5 | \( 1 + (-3.32 + 3.73i)T \) |
| 7 | \( 1 + (-2.29 - 6.61i)T \) |
good | 11 | \( 1 + 10.5iT - 121T^{2} \) |
| 13 | \( 1 + (14.9 - 14.9i)T - 169iT^{2} \) |
| 17 | \( 1 + (-15.4 + 15.4i)T - 289iT^{2} \) |
| 19 | \( 1 - 17.3T + 361T^{2} \) |
| 23 | \( 1 + (23.1 - 23.1i)T - 529iT^{2} \) |
| 29 | \( 1 + 23.7T + 841T^{2} \) |
| 31 | \( 1 - 33.1iT - 961T^{2} \) |
| 37 | \( 1 + (17.6 - 17.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 11.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (22.8 + 22.8i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-12.6 + 12.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (15.3 - 15.3i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 31.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (77.5 - 77.5i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 60.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (3.52 - 3.52i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 99.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-16.9 - 16.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 17.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-34.7 - 34.7i)T + 9.40e3iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.11264052374093650606752153897, −11.90859088430392423224733111629, −9.701523722122858600189743751466, −9.121918794822745563583504359824, −8.173143182064803712631872659136, −7.16129037930533689308516653195, −5.81132720171426277111054358937, −4.91432504637508061403853412932, −3.23226777643261535781669959012, −1.84102340375738935512747390263,
1.92117655662520040609441087917, 3.14080694764891585881946557829, 4.31346175412957920658504933852, 5.55371843717487531574627252143, 7.19885284160219830059417360094, 7.922648131575312438885260962491, 9.764570840723293864673026632766, 10.03650207636309261690724311120, 10.81690435962960183168345259303, 12.31053804589799192775248513909