L(s) = 1 | − 2-s + (0.420 − 1.68i)3-s + 4-s + (−1.95 − 1.08i)5-s + (−0.420 + 1.68i)6-s + (−2.37 − 1.16i)7-s − 8-s + (−2.64 − 1.41i)9-s + (1.95 + 1.08i)10-s + 2.82i·11-s + (0.420 − 1.68i)12-s − 0.841·13-s + (2.37 + 1.16i)14-s + (−2.64 + 2.82i)15-s + 16-s − 1.19i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.242 − 0.970i)3-s + 0.5·4-s + (−0.874 − 0.485i)5-s + (−0.171 + 0.685i)6-s + (−0.898 − 0.439i)7-s − 0.353·8-s + (−0.881 − 0.471i)9-s + (0.618 + 0.343i)10-s + 0.852i·11-s + (0.121 − 0.485i)12-s − 0.233·13-s + (0.635 + 0.311i)14-s + (−0.683 + 0.730i)15-s + 0.250·16-s − 0.288i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0876499 - 0.477162i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0876499 - 0.477162i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.420 + 1.68i)T \) |
| 5 | \( 1 + (1.95 + 1.08i)T \) |
| 7 | \( 1 + (2.37 + 1.16i)T \) |
good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 0.841T + 13T^{2} \) |
| 17 | \( 1 + 1.19iT - 17T^{2} \) |
| 19 | \( 1 + 4.55iT - 19T^{2} \) |
| 23 | \( 1 + 3.29T + 23T^{2} \) |
| 29 | \( 1 + 7.98iT - 29T^{2} \) |
| 31 | \( 1 + 5.53iT - 31T^{2} \) |
| 37 | \( 1 + 10.8iT - 37T^{2} \) |
| 41 | \( 1 - 7.82T + 41T^{2} \) |
| 43 | \( 1 - 4.65iT - 43T^{2} \) |
| 47 | \( 1 - 4.33iT - 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 3.91T + 59T^{2} \) |
| 61 | \( 1 - 10.0iT - 61T^{2} \) |
| 67 | \( 1 + 4.65iT - 67T^{2} \) |
| 71 | \( 1 + 12.6iT - 71T^{2} \) |
| 73 | \( 1 - 3.06T + 73T^{2} \) |
| 79 | \( 1 + 7.29T + 79T^{2} \) |
| 83 | \( 1 + 7.70iT - 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 8.11T + 97T^{2} \) |
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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
−12.02420267042255358636181873377, −11.11066877573098231187879158297, −9.705005802204891788305208645998, −8.956531284431379058380447728014, −7.65801762965544907626892232906, −7.29115884899780409439796462339, −6.05417945385030425722634806431, −4.14657155283287519386544003128, −2.51723553906575536474786011466, −0.48365635517016648534400069353,
2.92475385206497316195899961231, 3.80786978680849867275871315589, 5.59034028837580024410420999398, 6.79235097908172184367863594354, 8.167834709241382308367639119446, 8.796401453902772342643355396169, 9.982751956312556026890550486476, 10.59914020653738493442102167560, 11.60673368756340125306596684827, 12.46981942696961717509699283412