L(s) = 1 | + (1 − 1.73i)5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)11-s + (3 − 5.19i)13-s + 5·17-s + 7·19-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s + (−2 − 3.46i)29-s + (−3 + 5.19i)31-s − 1.99·35-s + 2·37-s + (1.5 − 2.59i)41-s + (−0.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (0.447 − 0.774i)5-s + (−0.188 − 0.327i)7-s + (−0.150 − 0.261i)11-s + (0.832 − 1.44i)13-s + 1.21·17-s + 1.60·19-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s + (−0.371 − 0.643i)29-s + (−0.538 + 0.933i)31-s − 0.338·35-s + 0.328·37-s + (0.234 − 0.405i)41-s + (−0.0762 − 0.132i)43-s + (−0.0714 + 0.123i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.114988660\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.114988660\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + (3 + 5.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8 + 13.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.508227197506560569408718308131, −7.81040410498050742052684919425, −7.27507507037195586287877188521, −5.92137996246263340333550092174, −5.60429072026148336553023111341, −4.89066051897363349659360383926, −3.53278829488311987689102791134, −3.16115408377711912018754264393, −1.53932410433059335798272424415, −0.75192659922573311748690149693,
1.29047280698067573522763717028, 2.35020843521276027703114615910, 3.24121997661626338696002781682, 4.06955222183206101322461703969, 5.17609183065731513297920291635, 5.94756986314824526750409431340, 6.58955870120543043287117378205, 7.31050518693202309473222622350, 8.086274102179368242786549265858, 9.019699575943128862377116581153