L(s) = 1 | + (−1 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (0.815 + 4.93i)5-s − 2.44·6-s + (5.39 − 5.39i)7-s + (2 + 2i)8-s + 2.99i·9-s + (−5.74 − 4.11i)10-s − 16.8·11-s + (2.44 − 2.44i)12-s + (−15.3 − 15.3i)13-s + 10.7i·14-s + (−5.04 + 7.04i)15-s − 4·16-s + (20.2 − 20.2i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (0.163 + 0.986i)5-s − 0.408·6-s + (0.771 − 0.771i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.574 − 0.411i)10-s − 1.53·11-s + (0.204 − 0.204i)12-s + (−1.17 − 1.17i)13-s + 0.771i·14-s + (−0.336 + 0.469i)15-s − 0.250·16-s + (1.19 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8290909102\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8290909102\) |
\(L(2)\) |
|
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 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 + (-0.815 - 4.93i)T \) |
| 23 | \( 1 + (3.39 + 3.39i)T \) |
good | 7 | \( 1 + (-5.39 + 5.39i)T - 49iT^{2} \) |
| 11 | \( 1 + 16.8T + 121T^{2} \) |
| 13 | \( 1 + (15.3 + 15.3i)T + 169iT^{2} \) |
| 17 | \( 1 + (-20.2 + 20.2i)T - 289iT^{2} \) |
| 19 | \( 1 + 27.2iT - 361T^{2} \) |
| 29 | \( 1 - 1.86iT - 841T^{2} \) |
| 31 | \( 1 + 56.4T + 961T^{2} \) |
| 37 | \( 1 + (-14.7 + 14.7i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 31.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-3.71 - 3.71i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (52.2 - 52.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (29.1 + 29.1i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 44.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 97.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-29.7 + 29.7i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 112.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (69.1 + 69.1i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 17.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-70.4 - 70.4i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 103. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (78.1 - 78.1i)T - 9.40e3iT^{2} \) |
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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
−10.02085063406732237879576266331, −9.475584564184020345095951505847, −7.948406193736448639586027361459, −7.67913793164934204041227364471, −6.97320348558480840820071242350, −5.34575527501025243607304791432, −4.96487049993800463939370241641, −3.23274399684737211487407673691, −2.37304949203816091890234665562, −0.31022779458824529116733912108,
1.62608490718976798519022694095, 2.19779937866165405632101206830, 3.73481407123118000600141298986, 5.04057879588942397436286278978, 5.73057621407843737244155650045, 7.35310920615138678682182453285, 8.158891788344957762170935959445, 8.490396977148700019641858159562, 9.636921499498462088854068027548, 10.14782894849648174826108438262