L(s) = 1 | + (0.468 + 0.811i)5-s + (0.5 − 0.866i)7-s + (−2.48 + 4.30i)11-s + (−0.622 − 1.07i)13-s − 5.22·17-s − 5.18·19-s + (1.00 + 1.73i)23-s + (2.06 − 3.57i)25-s + (3.43 − 5.95i)29-s + (−2.86 − 4.96i)31-s + 0.936·35-s + 9.73·37-s + (−5.73 − 9.93i)41-s + (4.80 − 8.31i)43-s + (0.984 − 1.70i)47-s + ⋯ |
L(s) = 1 | + (0.209 + 0.362i)5-s + (0.188 − 0.327i)7-s + (−0.749 + 1.29i)11-s + (−0.172 − 0.298i)13-s − 1.26·17-s − 1.18·19-s + (0.209 + 0.362i)23-s + (0.412 − 0.714i)25-s + (0.638 − 1.10i)29-s + (−0.514 − 0.891i)31-s + 0.158·35-s + 1.59·37-s + (−0.895 − 1.55i)41-s + (0.732 − 1.26i)43-s + (0.143 − 0.248i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0352 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0352 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.009100650\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.009100650\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.468 - 0.811i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.48 - 4.30i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.622 + 1.07i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 + (-1.00 - 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.43 + 5.95i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.86 + 4.96i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.73T + 37T^{2} \) |
| 41 | \( 1 + (5.73 + 9.93i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.80 + 8.31i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.984 + 1.70i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 + (2.43 + 4.22i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.52 + 2.63i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.573 + 0.994i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.83T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 + (6.05 - 10.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.431 + 0.747i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + (3.78 - 6.55i)T + (-48.5 - 84.0i)T^{2} \) |
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
−8.488797697611064746826436692835, −7.71094411348419740910180646373, −7.03876515679727747294600277989, −6.38663113118151937502202879478, −5.43791802697229777941055947239, −4.51101372210609641008341200861, −4.00928194266335608509735893802, −2.45190476060465821010526335905, −2.15988624856209357268234102952, −0.32073993572121578274539318166,
1.18032218913185701297972239413, 2.42044067809262258612937023955, 3.16139392887113656512095533396, 4.44059025150741623927248245996, 4.97265383325610532298923443803, 5.94953670691403483687515693956, 6.50507106175407695332710916929, 7.45351172802917856954116228687, 8.511134835690551988355593884589, 8.670923831557666570998135445745