| L(s) = 1 | − 2.82·5-s + 3.46·7-s − 11-s − 2.89·13-s + 0.635·17-s + 0.635·19-s + 4.89·23-s + 3.00·25-s − 6.29·29-s + 5.65·31-s − 9.79·35-s + 11.7·37-s − 6.29·41-s − 6.29·43-s − 0.898·47-s + 4.99·49-s − 4.09·53-s + 2.82·55-s + 5.79·59-s + 10.8·61-s + 8.19·65-s + 8.89·71-s + 6·73-s − 3.46·77-s − 4.73·79-s + 5.79·83-s − 1.79·85-s + ⋯ |
| L(s) = 1 | − 1.26·5-s + 1.30·7-s − 0.301·11-s − 0.804·13-s + 0.154·17-s + 0.145·19-s + 1.02·23-s + 0.600·25-s − 1.16·29-s + 1.01·31-s − 1.65·35-s + 1.93·37-s − 0.982·41-s − 0.959·43-s − 0.131·47-s + 0.714·49-s − 0.563·53-s + 0.381·55-s + 0.754·59-s + 1.39·61-s + 1.01·65-s + 1.05·71-s + 0.702·73-s − 0.394·77-s − 0.532·79-s + 0.636·83-s − 0.195·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.492553690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.492553690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 - 0.635T + 17T^{2} \) |
| 19 | \( 1 - 0.635T + 19T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 6.29T + 41T^{2} \) |
| 43 | \( 1 + 6.29T + 43T^{2} \) |
| 47 | \( 1 + 0.898T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 - 5.79T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 8.89T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 4.73T + 79T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.368032499501396845683948322217, −7.943518498802434992317503263963, −7.42992758432627514831461982775, −6.61003068966412737121765355911, −5.32694347678764857808983571835, −4.82791375944546040952834733745, −4.08335537641332017147662208604, −3.13009829133612173146307860230, −2.04868415354981813860767216608, −0.73943232627439840765552522754,
0.73943232627439840765552522754, 2.04868415354981813860767216608, 3.13009829133612173146307860230, 4.08335537641332017147662208604, 4.82791375944546040952834733745, 5.32694347678764857808983571835, 6.61003068966412737121765355911, 7.42992758432627514831461982775, 7.943518498802434992317503263963, 8.368032499501396845683948322217
