L(s) = 1 | + (0.536 + 0.929i)2-s + (1.21 − 2.10i)3-s + (0.424 − 0.734i)4-s + (−0.312 − 0.541i)5-s + 2.60·6-s + (1.21 + 2.34i)7-s + 3.05·8-s + (−1.45 − 2.52i)9-s + (0.335 − 0.581i)10-s + (0.354 − 0.613i)11-s + (−1.03 − 1.78i)12-s + (−1.53 + 2.39i)14-s − 1.52·15-s + (0.791 + 1.37i)16-s + (1.67 − 2.89i)17-s + (1.56 − 2.70i)18-s + ⋯ |
L(s) = 1 | + (0.379 + 0.657i)2-s + (0.701 − 1.21i)3-s + (0.212 − 0.367i)4-s + (−0.139 − 0.242i)5-s + 1.06·6-s + (0.459 + 0.888i)7-s + 1.08·8-s + (−0.485 − 0.840i)9-s + (0.106 − 0.183i)10-s + (0.106 − 0.185i)11-s + (−0.297 − 0.515i)12-s + (−0.409 + 0.638i)14-s − 0.392·15-s + (0.197 + 0.342i)16-s + (0.405 − 0.702i)17-s + (0.368 − 0.637i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.052876212\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.052876212\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.21 - 2.34i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.536 - 0.929i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.21 + 2.10i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.312 + 0.541i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.354 + 0.613i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.67 + 2.89i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.60 - 4.50i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.21 - 3.83i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6.59T + 29T^{2} \) |
| 31 | \( 1 + (-2.19 + 3.80i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.211 + 0.366i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.01T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + (4.03 + 6.99i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.348 + 0.603i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.93 - 8.54i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.34 + 4.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.21 - 9.02i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + (-2.54 + 4.40i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.95 - 3.38i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + (-6.68 - 11.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.202T + 97T^{2} \) |
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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
−9.460214123252139363325527975719, −8.536299514233707915233828816543, −7.84449350752575135104053610939, −7.29811844622274043367611504411, −6.39071617718141218865611012965, −5.58881494833954002748314146295, −4.84897513798660545473785596581, −3.32496550679887361235267047172, −2.11426438558131649232679157354, −1.28820719151761349354493384758,
1.62299104869001517069880590976, 3.12058999256617855564641355760, 3.45340777434853192627206789551, 4.50799481216731648343207802594, 4.95494942270776330946290846433, 6.71455252133643329108407672428, 7.49214142571656919691996733731, 8.301877536347153608494167140909, 9.166212619369674189515569746576, 10.07336371593836320111793743570