L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.499i)6-s + (−0.633 − 0.366i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−4.09 + 2.36i)11-s + 0.999·12-s + (−2.59 − 2.5i)13-s + 0.732·14-s + (−0.5 − 0.866i)16-s + (1.13 − 1.96i)17-s − 0.999i·18-s + (−1.09 − 0.633i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.353 − 0.204i)6-s + (−0.239 − 0.138i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−1.23 + 0.713i)11-s + 0.288·12-s + (−0.720 − 0.693i)13-s + 0.195·14-s + (−0.125 − 0.216i)16-s + (0.275 − 0.476i)17-s − 0.235i·18-s + (−0.251 − 0.145i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7941609029\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7941609029\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2.59 + 2.5i)T \) |
good | 7 | \( 1 + (0.633 + 0.366i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.09 - 2.36i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.13 + 1.96i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.09 + 0.633i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.09 - 5.36i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.23 + 2.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.46iT - 31T^{2} \) |
| 37 | \( 1 + (-9.06 + 5.23i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.86 + 5.69i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.83 - 6.63i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 8.19iT - 47T^{2} \) |
| 53 | \( 1 + 0.464T + 53T^{2} \) |
| 59 | \( 1 + (-6.92 - 4i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.598 - 1.03i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.63 + 5.56i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.09 + 0.633i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 9.73iT - 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 + (-2.19 + 1.26i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.19 + 3i)T + (48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.361457837363070875867155167040, −8.144409329188613570728642150960, −7.64643013507142960323129453040, −7.05695829845152019930208743135, −5.75631243232626555413781224685, −5.20613980985200007037442651216, −4.24680890881607784523993358335, −2.97652263717091875013959214311, −2.19564666206182251916709746146, −0.37364629654869154881557387663,
1.09577675853303949248725458752, 2.46784770139233999295301256328, 2.95367496971954104060929628859, 4.24027703156403770087322553962, 5.32376443054217428520377202082, 6.34677380575521993574943930300, 7.03416989196933358589694681604, 7.999473910311689566613891934275, 8.364905483913143251447901853029, 9.281476123051157153754284441784