| L(s) = 1 | + 2.28·2-s + 1.47·3-s + 3.22·4-s + 1.42·5-s + 3.36·6-s + 3.91·7-s + 2.80·8-s − 0.828·9-s + 3.26·10-s + 3.36·11-s + 4.75·12-s − 6.36·13-s + 8.95·14-s + 2.10·15-s − 0.0417·16-s + 5.56·17-s − 1.89·18-s − 5.42·19-s + 4.61·20-s + 5.76·21-s + 7.69·22-s − 4.62·23-s + 4.13·24-s − 2.95·25-s − 14.5·26-s − 5.64·27-s + 12.6·28-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 0.850·3-s + 1.61·4-s + 0.639·5-s + 1.37·6-s + 1.47·7-s + 0.991·8-s − 0.276·9-s + 1.03·10-s + 1.01·11-s + 1.37·12-s − 1.76·13-s + 2.39·14-s + 0.543·15-s − 0.0104·16-s + 1.34·17-s − 0.446·18-s − 1.24·19-s + 1.03·20-s + 1.25·21-s + 1.64·22-s − 0.964·23-s + 0.843·24-s − 0.591·25-s − 2.85·26-s − 1.08·27-s + 2.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.422620232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.422620232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1511 | \( 1 - T \) |
| good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 - 1.47T + 3T^{2} \) |
| 5 | \( 1 - 1.42T + 5T^{2} \) |
| 7 | \( 1 - 3.91T + 7T^{2} \) |
| 11 | \( 1 - 3.36T + 11T^{2} \) |
| 13 | \( 1 + 6.36T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 + 5.42T + 19T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 + 8.19T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 + 6.23T + 37T^{2} \) |
| 41 | \( 1 + 3.91T + 41T^{2} \) |
| 43 | \( 1 - 5.53T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 6.01T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 5.61T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 2.33T + 73T^{2} \) |
| 79 | \( 1 - 6.37T + 79T^{2} \) |
| 83 | \( 1 - 2.84T + 83T^{2} \) |
| 89 | \( 1 - 1.12T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−9.462237239170318307948752890836, −8.579271984854064561948658995454, −7.72129516919817970543807631080, −6.99052610676925715448660186079, −5.71111590793048996928210192291, −5.39947983419390902871801983550, −4.30295333341790195968077683376, −3.69044199027600385679252075939, −2.32399680940714294283455097978, −1.98468425058471206596513974640,
1.98468425058471206596513974640, 2.32399680940714294283455097978, 3.69044199027600385679252075939, 4.30295333341790195968077683376, 5.39947983419390902871801983550, 5.71111590793048996928210192291, 6.99052610676925715448660186079, 7.72129516919817970543807631080, 8.579271984854064561948658995454, 9.462237239170318307948752890836
