L(s) = 1 | + (−2.54 + 1.59i)3-s + (4.96 − 0.625i)5-s + 5.93i·7-s + (3.92 − 8.10i)9-s + 5.99·11-s − 15.6·13-s + (−11.6 + 9.49i)15-s − 14.2·17-s + 29.6i·19-s + (−9.46 − 15.0i)21-s + 21.5·23-s + (24.2 − 6.20i)25-s + (2.93 + 26.8i)27-s − 18.1·29-s − 8.55·31-s + ⋯ |
L(s) = 1 | + (−0.847 + 0.531i)3-s + (0.992 − 0.125i)5-s + 0.848i·7-s + (0.435 − 0.900i)9-s + 0.545·11-s − 1.20·13-s + (−0.774 + 0.632i)15-s − 0.836·17-s + 1.55i·19-s + (−0.450 − 0.718i)21-s + 0.935·23-s + (0.968 − 0.248i)25-s + (0.108 + 0.994i)27-s − 0.625·29-s − 0.276·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.023623948\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.023623948\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.54 - 1.59i)T \) |
| 5 | \( 1 + (-4.96 + 0.625i)T \) |
good | 7 | \( 1 - 5.93iT - 49T^{2} \) |
| 11 | \( 1 - 5.99T + 121T^{2} \) |
| 13 | \( 1 + 15.6T + 169T^{2} \) |
| 17 | \( 1 + 14.2T + 289T^{2} \) |
| 19 | \( 1 - 29.6iT - 361T^{2} \) |
| 23 | \( 1 - 21.5T + 529T^{2} \) |
| 29 | \( 1 + 18.1T + 841T^{2} \) |
| 31 | \( 1 + 8.55T + 961T^{2} \) |
| 37 | \( 1 - 13.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 18.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 27.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 37.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 93.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 83.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 39.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 103.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 14.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 95.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 133.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 116. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 19.9iT - 9.40e3T^{2} \) |
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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
−10.20497970515686934044310661363, −9.223867011383437379167808113333, −9.039805139142321351794936055557, −7.49197852067328663232776497456, −6.47283748404301877339141068181, −5.77980350202355170380955810771, −5.11806203132785938993137862914, −4.14216111498512435012614019921, −2.69768714972217950781086425557, −1.48358072667737696907090606800,
0.36019654994395228533000361591, 1.64693350926505189484544823108, 2.75762815270924699490350470051, 4.48226563742581649009649133492, 5.09783117852339945283602157798, 6.18349431789382123925235316959, 7.01030283660653251164966469497, 7.32006862375960349287108756308, 8.824756638029619825369212101084, 9.607938072662575390912475746625