L(s) = 1 | + (−2.13 + 1.55i)2-s + (0.309 + 0.951i)3-s + (1.53 − 4.72i)4-s + (−2.49 − 1.81i)5-s + (−2.13 − 1.55i)6-s + (−0.309 + 0.951i)7-s + (2.42 + 7.45i)8-s + (−0.809 + 0.587i)9-s + 8.14·10-s + (1.45 − 2.98i)11-s + 4.96·12-s + (4.56 − 3.31i)13-s + (−0.815 − 2.51i)14-s + (0.952 − 2.93i)15-s + (−8.70 − 6.32i)16-s + (4.53 + 3.29i)17-s + ⋯ |
L(s) = 1 | + (−1.51 + 1.09i)2-s + (0.178 + 0.549i)3-s + (0.767 − 2.36i)4-s + (−1.11 − 0.810i)5-s + (−0.871 − 0.633i)6-s + (−0.116 + 0.359i)7-s + (0.856 + 2.63i)8-s + (−0.269 + 0.195i)9-s + 2.57·10-s + (0.438 − 0.898i)11-s + 1.43·12-s + (1.26 − 0.919i)13-s + (−0.218 − 0.671i)14-s + (0.246 − 0.757i)15-s + (−2.17 − 1.58i)16-s + (1.09 + 0.798i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.512967 + 0.121304i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.512967 + 0.121304i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (-1.45 + 2.98i)T \) |
good | 2 | \( 1 + (2.13 - 1.55i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (2.49 + 1.81i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (-4.56 + 3.31i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.53 - 3.29i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.0379 - 0.116i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 1.67T + 23T^{2} \) |
| 29 | \( 1 + (-2.73 + 8.41i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.29 - 0.939i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.28 + 3.96i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.07 - 3.31i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 5.10T + 43T^{2} \) |
| 47 | \( 1 + (0.492 + 1.51i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-9.59 + 6.96i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.04 + 6.29i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-6.86 - 4.98i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 8.04T + 67T^{2} \) |
| 71 | \( 1 + (5.05 + 3.67i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.08 + 3.33i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.60 + 5.52i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-7.63 - 5.54i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 8.45T + 89T^{2} \) |
| 97 | \( 1 + (-4.59 + 3.34i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.89159391159871567737139829082, −10.97059833308189740513731836950, −10.05324692858186243747291145363, −8.941120641119414616913293502805, −8.312391619215349221807462984115, −7.87821001317180372383760782212, −6.25733512997718449042122382963, −5.41054223561132007014542998328, −3.70488770365727301035004920386, −0.842098642116589492520257484617,
1.31832354572488906208429765563, 3.00304454966175892501691603821, 3.93056104599807862193991764083, 6.89552068778776400722821010413, 7.28598370046274693102612940037, 8.331591309487685499706779265986, 9.225564681744568341300175140995, 10.29166033073565609930860736452, 11.18171607577897985887346456924, 11.79589673510884719457172543025