| L(s) = 1 | + 0.118·2-s + 3-s − 1.98·4-s + 2.39·5-s + 0.118·6-s + 2.97·7-s − 0.470·8-s + 9-s + 0.282·10-s − 11-s − 1.98·12-s + 0.350·14-s + 2.39·15-s + 3.91·16-s + 4.58·17-s + 0.118·18-s + 4.85·19-s − 4.76·20-s + 2.97·21-s − 0.118·22-s + 1.75·23-s − 0.470·24-s + 0.748·25-s + 27-s − 5.89·28-s + 5.42·29-s + 0.282·30-s + ⋯ |
| L(s) = 1 | + 0.0834·2-s + 0.577·3-s − 0.993·4-s + 1.07·5-s + 0.0481·6-s + 1.12·7-s − 0.166·8-s + 0.333·9-s + 0.0894·10-s − 0.301·11-s − 0.573·12-s + 0.0937·14-s + 0.619·15-s + 0.979·16-s + 1.11·17-s + 0.0278·18-s + 1.11·19-s − 1.06·20-s + 0.648·21-s − 0.0251·22-s + 0.365·23-s − 0.0960·24-s + 0.149·25-s + 0.192·27-s − 1.11·28-s + 1.00·29-s + 0.0516·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.222756277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.222756277\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.118T + 2T^{2} \) |
| 5 | \( 1 - 2.39T + 5T^{2} \) |
| 7 | \( 1 - 2.97T + 7T^{2} \) |
| 17 | \( 1 - 4.58T + 17T^{2} \) |
| 19 | \( 1 - 4.85T + 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 - 1.91T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 + 5.41T + 41T^{2} \) |
| 43 | \( 1 + 7.01T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 4.41T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 - 5.44T + 61T^{2} \) |
| 67 | \( 1 + 5.70T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 2.55T + 79T^{2} \) |
| 83 | \( 1 - 9.41T + 83T^{2} \) |
| 89 | \( 1 + 0.0288T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.098717087484846841367721020848, −7.80626299800105682177975077396, −6.71294925967913184155625356039, −5.76557456852392872963505967915, −5.08240347726257144598755091786, −4.76323956881985050536612233676, −3.59836364335862309837260123482, −2.89197858814715921435788803213, −1.75893949475918735599791464553, −1.02556919575047258981107420566,
1.02556919575047258981107420566, 1.75893949475918735599791464553, 2.89197858814715921435788803213, 3.59836364335862309837260123482, 4.76323956881985050536612233676, 5.08240347726257144598755091786, 5.76557456852392872963505967915, 6.71294925967913184155625356039, 7.80626299800105682177975077396, 8.098717087484846841367721020848
