L(s) = 1 | + (−1.5 + 0.866i)2-s + (1 + 1.73i)3-s + (0.5 − 0.866i)4-s − 1.73i·5-s + (−3 − 1.73i)6-s − 1.73i·8-s + (−0.499 + 0.866i)9-s + (1.49 + 2.59i)10-s + 2·12-s + (2.5 − 2.59i)13-s + (2.99 − 1.73i)15-s + (2.49 + 4.33i)16-s + (−1.5 + 2.59i)17-s − 1.73i·18-s + (3 + 1.73i)19-s + (−1.50 − 0.866i)20-s + ⋯ |
L(s) = 1 | + (−1.06 + 0.612i)2-s + (0.577 + 0.999i)3-s + (0.250 − 0.433i)4-s − 0.774i·5-s + (−1.22 − 0.707i)6-s − 0.612i·8-s + (−0.166 + 0.288i)9-s + (0.474 + 0.821i)10-s + 0.577·12-s + (0.693 − 0.720i)13-s + (0.774 − 0.447i)15-s + (0.624 + 1.08i)16-s + (−0.363 + 0.630i)17-s − 0.408i·18-s + (0.688 + 0.397i)19-s + (−0.335 − 0.193i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.840982 + 0.649602i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.840982 + 0.649602i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.5 + 2.59i)T \) |
good | 2 | \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (-7.5 + 4.33i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (6 + 3.46i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3 + 1.73i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6 + 3.46i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.35634903222647307214249563420, −9.523042953568066935061081759641, −9.093346863751488825405663314713, −8.312282453542354675124519549229, −7.67708443227877586900418878810, −6.42761798679709011725480848724, −5.31126613198605125588421274489, −4.12940735671574564333337824063, −3.28612597031454256910109698669, −1.12344547484835562828742641585,
1.05120114925961720351497648493, 2.27433639748080715438872033636, 3.03066183000832866525146587183, 4.78070743596815260235133973597, 6.32759656866695939344260748572, 7.12105337510844211703819790809, 7.87655938772660249829676498002, 8.784606032870381705088067350927, 9.368023433848938627261473378530, 10.45077125195733125310922020347