| L(s) = 1 | + 2.45·2-s + 4.04·4-s − 3.82·5-s − 0.0899·7-s + 5.01·8-s − 9.40·10-s − 11-s − 0.000700·13-s − 0.221·14-s + 4.24·16-s + 1.57·17-s + 6.57·19-s − 15.4·20-s − 2.45·22-s − 6.23·23-s + 9.63·25-s − 0.00172·26-s − 0.363·28-s + 6.55·29-s − 3.63·31-s + 0.401·32-s + 3.87·34-s + 0.344·35-s + 10.7·37-s + 16.1·38-s − 19.1·40-s + 0.623·41-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 2.02·4-s − 1.71·5-s − 0.0340·7-s + 1.77·8-s − 2.97·10-s − 0.301·11-s − 0.000194·13-s − 0.0590·14-s + 1.06·16-s + 0.382·17-s + 1.50·19-s − 3.45·20-s − 0.523·22-s − 1.30·23-s + 1.92·25-s − 0.000337·26-s − 0.0686·28-s + 1.21·29-s − 0.653·31-s + 0.0710·32-s + 0.665·34-s + 0.0581·35-s + 1.76·37-s + 2.62·38-s − 3.03·40-s + 0.0974·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6633 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.205317701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.205317701\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 7 | \( 1 + 0.0899T + 7T^{2} \) |
| 13 | \( 1 + 0.000700T + 13T^{2} \) |
| 17 | \( 1 - 1.57T + 17T^{2} \) |
| 19 | \( 1 - 6.57T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 + 3.63T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 0.623T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 + 0.234T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 71 | \( 1 - 9.43T + 71T^{2} \) |
| 73 | \( 1 - 9.58T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 2.80T + 83T^{2} \) |
| 89 | \( 1 + 6.88T + 89T^{2} \) |
| 97 | \( 1 - 0.252T + 97T^{2} \) |
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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
−7.73932226933643833786148340744, −7.26809724405799519565270140715, −6.43356367235363160537118794022, −5.70896092379783925321357029205, −4.91617579606904481339032301200, −4.38360226975561133201125500725, −3.63807026780394825353819458897, −3.21978982074133868485931850646, −2.31504811789574743726703377185, −0.801816679530657164551859742651,
0.801816679530657164551859742651, 2.31504811789574743726703377185, 3.21978982074133868485931850646, 3.63807026780394825353819458897, 4.38360226975561133201125500725, 4.91617579606904481339032301200, 5.70896092379783925321357029205, 6.43356367235363160537118794022, 7.26809724405799519565270140715, 7.73932226933643833786148340744
