L(s) = 1 | + (−0.5 − 0.866i)7-s + (3 + 5.19i)11-s + (−1 + 1.73i)13-s − 4·19-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s + (−3 − 5.19i)29-s + (−4 + 6.92i)31-s + 2·37-s + (−6 + 10.3i)41-s + (2 + 3.46i)43-s + (−6 − 10.3i)47-s + (−0.499 + 0.866i)49-s − 6·53-s + (5 + 8.66i)61-s + ⋯ |
L(s) = 1 | + (−0.188 − 0.327i)7-s + (0.904 + 1.56i)11-s + (−0.277 + 0.480i)13-s − 0.917·19-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s + (−0.557 − 0.964i)29-s + (−0.718 + 1.24i)31-s + 0.328·37-s + (−0.937 + 1.62i)41-s + (0.304 + 0.528i)43-s + (−0.875 − 1.51i)47-s + (−0.0714 + 0.123i)49-s − 0.824·53-s + (0.640 + 1.10i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.264580172\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.264580172\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (6 - 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.277514125400977374924380205898, −8.581633392470210362959985492589, −7.55487425682306110698043040046, −6.80280541089586687415480976047, −6.45421799154760728205789787926, −5.04670819917669227594888150587, −4.46884447186603265294106039120, −3.61902159725980669399051113580, −2.36346957374264415656614906676, −1.39522684320367510089473238635,
0.43932082541865337390556094970, 1.83537165811994897304706491537, 3.11662560351249626213377973979, 3.71241073276804783745447111211, 4.87651503975233622159056832727, 5.80930879213329643422588999269, 6.31189553496052650422038856896, 7.26718065801961995198100705756, 8.139114171419124658855805662152, 8.916673708942122618435374206025