| L(s) = 1 | − 0.991·2-s − 5.36·3-s − 7.01·4-s − 18.5·5-s + 5.31·6-s + 13.4·7-s + 14.8·8-s + 1.74·9-s + 18.4·10-s − 11·11-s + 37.6·12-s − 13.2·14-s + 99.6·15-s + 41.3·16-s + 7.02·17-s − 1.72·18-s − 2.93·19-s + 130.·20-s − 71.8·21-s + 10.9·22-s − 62.7·23-s − 79.8·24-s + 220.·25-s + 135.·27-s − 94.0·28-s − 111.·29-s − 98.8·30-s + ⋯ |
| L(s) = 1 | − 0.350·2-s − 1.03·3-s − 0.877·4-s − 1.66·5-s + 0.361·6-s + 0.723·7-s + 0.658·8-s + 0.0644·9-s + 0.583·10-s − 0.301·11-s + 0.904·12-s − 0.253·14-s + 1.71·15-s + 0.646·16-s + 0.100·17-s − 0.0226·18-s − 0.0354·19-s + 1.45·20-s − 0.746·21-s + 0.105·22-s − 0.568·23-s − 0.679·24-s + 1.76·25-s + 0.965·27-s − 0.634·28-s − 0.712·29-s − 0.601·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1436938520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1436938520\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.991T + 8T^{2} \) |
| 3 | \( 1 + 5.36T + 27T^{2} \) |
| 5 | \( 1 + 18.5T + 125T^{2} \) |
| 7 | \( 1 - 13.4T + 343T^{2} \) |
| 17 | \( 1 - 7.02T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.93T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 111.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 62.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 17.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 436.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 595.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 745.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 443.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 266.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 469.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 129.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 311.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 553.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 557.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 270.T + 9.12e5T^{2} \) |
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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
−8.677725460939567657638145291057, −7.989278265172180650851758555378, −7.66993069050063408542133432302, −6.52881295387106269961842746307, −5.45718823552238053123823703454, −4.73314550497753085471632875420, −4.19689751312248343684305531854, −3.16481942932928044698020829167, −1.33542630170129152520547781194, −0.22042508724246486479279787372,
0.22042508724246486479279787372, 1.33542630170129152520547781194, 3.16481942932928044698020829167, 4.19689751312248343684305531854, 4.73314550497753085471632875420, 5.45718823552238053123823703454, 6.52881295387106269961842746307, 7.66993069050063408542133432302, 7.989278265172180650851758555378, 8.677725460939567657638145291057
