L(s) = 1 | − 0.0795·2-s − 1.99·4-s − 3.50·5-s + 0.586·7-s + 0.317·8-s + 0.278·10-s + 4.15·11-s + 7.11·13-s − 0.0466·14-s + 3.96·16-s − 2.23·17-s + 3.72·19-s + 6.97·20-s − 0.330·22-s + 23-s + 7.25·25-s − 0.565·26-s − 1.16·28-s − 29-s − 0.517·31-s − 0.950·32-s + 0.177·34-s − 2.05·35-s + 8.76·37-s − 0.296·38-s − 1.11·40-s + 0.620·41-s + ⋯ |
L(s) = 1 | − 0.0562·2-s − 0.996·4-s − 1.56·5-s + 0.221·7-s + 0.112·8-s + 0.0880·10-s + 1.25·11-s + 1.97·13-s − 0.0124·14-s + 0.990·16-s − 0.540·17-s + 0.855·19-s + 1.56·20-s − 0.0703·22-s + 0.208·23-s + 1.45·25-s − 0.110·26-s − 0.221·28-s − 0.185·29-s − 0.0928·31-s − 0.167·32-s + 0.0304·34-s − 0.347·35-s + 1.44·37-s − 0.0481·38-s − 0.175·40-s + 0.0969·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.284273944\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.284273944\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 0.0795T + 2T^{2} \) |
| 5 | \( 1 + 3.50T + 5T^{2} \) |
| 7 | \( 1 - 0.586T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 - 7.11T + 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 - 3.72T + 19T^{2} \) |
| 31 | \( 1 + 0.517T + 31T^{2} \) |
| 37 | \( 1 - 8.76T + 37T^{2} \) |
| 41 | \( 1 - 0.620T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 + 9.76T + 47T^{2} \) |
| 53 | \( 1 - 3.66T + 53T^{2} \) |
| 59 | \( 1 - 5.72T + 59T^{2} \) |
| 61 | \( 1 - 0.946T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 1.43T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 0.300T + 79T^{2} \) |
| 83 | \( 1 + 8.47T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.213929628149711749263691420096, −7.58692497263154608149559108455, −6.67341130841897106522248166010, −6.02293433855732119953798299578, −4.95072862891236013677226603409, −4.31141148445396370232472289250, −3.66274306153327653871434448635, −3.33119940219019253381477439056, −1.45030832675820530636694228616, −0.68181155435695067088032305787,
0.68181155435695067088032305787, 1.45030832675820530636694228616, 3.33119940219019253381477439056, 3.66274306153327653871434448635, 4.31141148445396370232472289250, 4.95072862891236013677226603409, 6.02293433855732119953798299578, 6.67341130841897106522248166010, 7.58692497263154608149559108455, 8.213929628149711749263691420096