L(s) = 1 | + (2.41 + 2.41i)3-s + (2.54 − 2.54i)5-s − i·7-s + 8.70i·9-s + (−0.764 + 0.764i)11-s + (1.26 + 1.26i)13-s + 12.2·15-s + 5.65·17-s + (−0.0445 − 0.0445i)19-s + (2.41 − 2.41i)21-s + 1.46i·23-s − 7.91i·25-s + (−13.8 + 13.8i)27-s + (−3.56 − 3.56i)29-s − 4.75·31-s + ⋯ |
L(s) = 1 | + (1.39 + 1.39i)3-s + (1.13 − 1.13i)5-s − 0.377i·7-s + 2.90i·9-s + (−0.230 + 0.230i)11-s + (0.351 + 0.351i)13-s + 3.17·15-s + 1.37·17-s + (−0.0102 − 0.0102i)19-s + (0.527 − 0.527i)21-s + 0.305i·23-s − 1.58i·25-s + (−2.65 + 2.65i)27-s + (−0.662 − 0.662i)29-s − 0.853·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.546223442\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.546223442\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (-2.41 - 2.41i)T + 3iT^{2} \) |
| 5 | \( 1 + (-2.54 + 2.54i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.764 - 0.764i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.26 - 1.26i)T + 13iT^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + (0.0445 + 0.0445i)T + 19iT^{2} \) |
| 23 | \( 1 - 1.46iT - 23T^{2} \) |
| 29 | \( 1 + (3.56 + 3.56i)T + 29iT^{2} \) |
| 31 | \( 1 + 4.75T + 31T^{2} \) |
| 37 | \( 1 + (-5.09 + 5.09i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.50iT - 41T^{2} \) |
| 43 | \( 1 + (3.22 - 3.22i)T - 43iT^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 + (4.66 - 4.66i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.38 + 5.38i)T - 59iT^{2} \) |
| 61 | \( 1 + (-6.80 - 6.80i)T + 61iT^{2} \) |
| 67 | \( 1 + (4.92 + 4.92i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.19iT - 71T^{2} \) |
| 73 | \( 1 - 8.59iT - 73T^{2} \) |
| 79 | \( 1 + 7.84T + 79T^{2} \) |
| 83 | \( 1 + (-7.43 - 7.43i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.32iT - 89T^{2} \) |
| 97 | \( 1 - 0.485T + 97T^{2} \) |
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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
−9.429371755160350842399753397308, −8.857981051923589402924193904953, −8.097715125293779897485086663869, −7.38771029718275050504019307348, −5.71682773549374282549057761531, −5.25556907236541639563323246027, −4.30686024023124319659380086002, −3.65237926572438844411237821619, −2.48820749219019840950204638049, −1.55637878589169235210736494083,
1.30631590201103069961040754490, 2.18917315472997920411263295558, 3.00216051065976483223267421149, 3.49650289489358980301531696290, 5.50155742579546223738728438106, 6.22075821792148183856255572946, 6.82201834128959239570166869903, 7.64910874446617739741783881151, 8.211972598712816684963686160630, 9.136169712720462317802173013868