L(s) = 1 | − 1.20·2-s − 0.559·4-s + 3.40·5-s + 4.71·7-s + 3.07·8-s − 4.08·10-s + 1.68·11-s − 2.33·13-s − 5.65·14-s − 2.56·16-s + 4.01·17-s + 0.666·19-s − 1.90·20-s − 2.02·22-s − 23-s + 6.56·25-s + 2.80·26-s − 2.63·28-s + 29-s − 4.38·31-s − 3.06·32-s − 4.82·34-s + 16.0·35-s + 1.15·37-s − 0.799·38-s + 10.4·40-s − 10.3·41-s + ⋯ |
L(s) = 1 | − 0.848·2-s − 0.279·4-s + 1.52·5-s + 1.78·7-s + 1.08·8-s − 1.29·10-s + 0.508·11-s − 0.647·13-s − 1.51·14-s − 0.641·16-s + 0.974·17-s + 0.152·19-s − 0.425·20-s − 0.431·22-s − 0.208·23-s + 1.31·25-s + 0.549·26-s − 0.498·28-s + 0.185·29-s − 0.787·31-s − 0.541·32-s − 0.826·34-s + 2.70·35-s + 0.189·37-s − 0.129·38-s + 1.65·40-s − 1.61·41-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
\(L(1)\) |
\(\approx\) |
\(2.139252688\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.139252688\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.20T + 2T^{2} \) |
| 5 | \( 1 - 3.40T + 5T^{2} \) |
| 7 | \( 1 - 4.71T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 + 2.33T + 13T^{2} \) |
| 17 | \( 1 - 4.01T + 17T^{2} \) |
| 19 | \( 1 - 0.666T + 19T^{2} \) |
| 31 | \( 1 + 4.38T + 31T^{2} \) |
| 37 | \( 1 - 1.15T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4.96T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 7.60T + 53T^{2} \) |
| 59 | \( 1 + 2.67T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 7.64T + 67T^{2} \) |
| 71 | \( 1 - 2.23T + 71T^{2} \) |
| 73 | \( 1 - 3.18T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 0.691T + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.171255552650849976908583369674, −7.58686276278185413819890440827, −6.86198399775542412172890018943, −5.79840585729662181128448774763, −5.16109517205134644231195978966, −4.76530839606022147072469804684, −3.68878763896000204630180369162, −2.24310017539424520863876629531, −1.70238692860697245978621194745, −0.973611190490846696448294376533,
0.973611190490846696448294376533, 1.70238692860697245978621194745, 2.24310017539424520863876629531, 3.68878763896000204630180369162, 4.76530839606022147072469804684, 5.16109517205134644231195978966, 5.79840585729662181128448774763, 6.86198399775542412172890018943, 7.58686276278185413819890440827, 8.171255552650849976908583369674