L(s) = 1 | + i·2-s + (−0.618 − 1.61i)3-s − 4-s − 5-s + (1.61 − 0.618i)6-s + (0.381 − 2.61i)7-s − i·8-s + (−2.23 + 2.00i)9-s − i·10-s − 4.47i·11-s + (0.618 + 1.61i)12-s − 3.23i·13-s + (2.61 + 0.381i)14-s + (0.618 + 1.61i)15-s + 16-s − 0.763·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.356 − 0.934i)3-s − 0.5·4-s − 0.447·5-s + (0.660 − 0.252i)6-s + (0.144 − 0.989i)7-s − 0.353i·8-s + (−0.745 + 0.666i)9-s − 0.316i·10-s − 1.34i·11-s + (0.178 + 0.467i)12-s − 0.897i·13-s + (0.699 + 0.102i)14-s + (0.159 + 0.417i)15-s + 0.250·16-s − 0.185·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.629388 - 0.504195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.629388 - 0.504195i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.618 + 1.61i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.381 + 2.61i)T \) |
good | 11 | \( 1 + 4.47iT - 11T^{2} \) |
| 13 | \( 1 + 3.23iT - 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 - 0.472iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 5.70iT - 29T^{2} \) |
| 31 | \( 1 - 7.23iT - 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 + 8.47iT - 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 2.76iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 2.76iT - 71T^{2} \) |
| 73 | \( 1 - 6.76iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 5.23iT - 97T^{2} \) |
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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
−12.37981067120107325600304814535, −11.12354476211439981238924812117, −10.49285607584865088829789938026, −8.688746073863802366844478885316, −7.985756064625003225791222056715, −7.08277001978161789814923756476, −6.13616947239813470342391372262, −4.96317959414577807808508999949, −3.33634467327170917169484353562, −0.74380502191312902225374957915,
2.36425259544367887198175546683, 4.01044565414653378996205278864, 4.83596773642701562228480550864, 6.11185740583984162563712274414, 7.74747172114191661064883955424, 9.164419542776622853630167849868, 9.547487619612675640524745244220, 10.74325855418097507554286972674, 11.79032693231415015685817541672, 12.01840924437992871122136638973