L(s) = 1 | − 1.73i·3-s + 3.58·5-s − 2.99·9-s − 0.913i·11-s + 1.16·13-s − 6.20i·15-s + 26.7·17-s + 17.5i·19-s − 27.1i·23-s − 12.1·25-s + 5.19i·27-s − 2·29-s + 45.6i·31-s − 1.58·33-s + 47.4·37-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.716·5-s − 0.333·9-s − 0.0830i·11-s + 0.0896·13-s − 0.413i·15-s + 1.57·17-s + 0.921i·19-s − 1.18i·23-s − 0.486·25-s + 0.192i·27-s − 0.0689·29-s + 1.47i·31-s − 0.0479·33-s + 1.28·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.541387084\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.541387084\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.58T + 25T^{2} \) |
| 11 | \( 1 + 0.913iT - 121T^{2} \) |
| 13 | \( 1 - 1.16T + 169T^{2} \) |
| 17 | \( 1 - 26.7T + 289T^{2} \) |
| 19 | \( 1 - 17.5iT - 361T^{2} \) |
| 23 | \( 1 + 27.1iT - 529T^{2} \) |
| 29 | \( 1 + 2T + 841T^{2} \) |
| 31 | \( 1 - 45.6iT - 961T^{2} \) |
| 37 | \( 1 - 47.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 42.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 14.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 8.37iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 41.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 27.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 11.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 71.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 55.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 95.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 63.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 32.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 120.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 107.T + 9.40e3T^{2} \) |
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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
−8.620365919696308720012113843923, −7.965841727010312950898190494468, −7.24822432752047732128056161669, −6.24666005125534953115223614269, −5.81864141987371806234379322813, −4.93125317905845821820078620473, −3.74239688550247573896395756395, −2.79287603186568256330264325231, −1.78520858896390217917524749458, −0.841891824488564857182415727780,
0.854620506712297607983094383055, 2.11968399458247026660151130961, 3.11450493761426200643419194707, 4.00906684809652610749078244715, 4.98270140691602400346375030752, 5.74077286955525622593071484470, 6.27825449971452879222982162435, 7.55426893745505545960817753178, 7.961238052947218698267774200906, 9.266180642057986071759285184531