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 | \( 1 + (-34.7 + 34.7i)T - 9.40e3iT^{2} \) |
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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
−12.31053804589799192775248513909, −10.81690435962960183168345259303, −10.03650207636309261690724311120, −9.764570840723293864673026632766, −7.922648131575312438885260962491, −7.19885284160219830059417360094, −5.55371843717487531574627252143, −4.31346175412957920658504933852, −3.14080694764891585881946557829, −1.92117655662520040609441087917,
1.84102340375738935512747390263, 3.23226777643261535781669959012, 4.91432504637508061403853412932, 5.81132720171426277111054358937, 7.16129037930533689308516653195, 8.173143182064803712631872659136, 9.121918794822745563583504359824, 9.701523722122858600189743751466, 11.90859088430392423224733111629, 12.11264052374093650606752153897