L(s) = 1 | + (0.796 + 1.53i)3-s + (−1.02 + 1.77i)5-s + (−0.5 − 0.866i)7-s + (−1.73 + 2.45i)9-s + (2.52 + 4.37i)11-s + (0.5 − 0.866i)13-s + (−3.55 − 0.162i)15-s + 0.273·17-s − 5.38·19-s + (0.933 − 1.45i)21-s + (2.66 − 4.61i)23-s + (0.390 + 0.676i)25-s + (−5.14 − 0.708i)27-s + (4.16 + 7.21i)29-s + (5.08 − 8.80i)31-s + ⋯ |
L(s) = 1 | + (0.460 + 0.887i)3-s + (−0.459 + 0.795i)5-s + (−0.188 − 0.327i)7-s + (−0.576 + 0.816i)9-s + (0.761 + 1.31i)11-s + (0.138 − 0.240i)13-s + (−0.917 − 0.0418i)15-s + 0.0662·17-s − 1.23·19-s + (0.203 − 0.318i)21-s + (0.555 − 0.962i)23-s + (0.0780 + 0.135i)25-s + (−0.990 − 0.136i)27-s + (0.773 + 1.33i)29-s + (0.912 − 1.58i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.867887 + 0.943451i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.867887 + 0.943451i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.796 - 1.53i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (1.02 - 1.77i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.52 - 4.37i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 0.273T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 + (-2.66 + 4.61i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.16 - 7.21i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.08 + 8.80i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.16T + 37T^{2} \) |
| 41 | \( 1 + (-2.52 + 4.37i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.30 + 3.98i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.690 + 1.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 3.43T + 53T^{2} \) |
| 59 | \( 1 + (0.890 - 1.54i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.390 + 0.676i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.19 + 7.25i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.78T + 71T^{2} \) |
| 73 | \( 1 - 9.38T + 73T^{2} \) |
| 79 | \( 1 + (6.47 + 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.86 - 4.95i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + (-1.10 - 1.92i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.22196548863588868497500324697, −11.05349493708087182387082518610, −10.41929137928389158499989517274, −9.535085168137362020623584938653, −8.507725440296463741338428054880, −7.36730083087809309378885372718, −6.41679944726593406557584588540, −4.68399406015828843288797991777, −3.86315390451844595562142974679, −2.54449787003725554882635162276,
1.06761587383282006321810434411, 2.92378284490732128024108184440, 4.27846002097281827006952561126, 5.91962011969041426407445239974, 6.72762418603509454033685444281, 8.241324239741766782229839783744, 8.534774097398416265791193832353, 9.563441597851516902758463293679, 11.20528713869799952842530977468, 11.90170491411507061163486125505