L(s) = 1 | + 2-s + 1.56·3-s + 4-s + 4.16·5-s + 1.56·6-s + 1.53·7-s + 8-s − 0.543·9-s + 4.16·10-s − 3.34·11-s + 1.56·12-s + 6.13·13-s + 1.53·14-s + 6.53·15-s + 16-s − 1.87·17-s − 0.543·18-s + 0.663·19-s + 4.16·20-s + 2.40·21-s − 3.34·22-s + 4.67·23-s + 1.56·24-s + 12.3·25-s + 6.13·26-s − 5.55·27-s + 1.53·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.904·3-s + 0.5·4-s + 1.86·5-s + 0.639·6-s + 0.580·7-s + 0.353·8-s − 0.181·9-s + 1.31·10-s − 1.00·11-s + 0.452·12-s + 1.70·13-s + 0.410·14-s + 1.68·15-s + 0.250·16-s − 0.453·17-s − 0.128·18-s + 0.152·19-s + 0.932·20-s + 0.525·21-s − 0.712·22-s + 0.974·23-s + 0.319·24-s + 2.47·25-s + 1.20·26-s − 1.06·27-s + 0.290·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.858796130\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.858796130\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 13 | \( 1 - 6.13T + 13T^{2} \) |
| 17 | \( 1 + 1.87T + 17T^{2} \) |
| 19 | \( 1 - 0.663T + 19T^{2} \) |
| 23 | \( 1 - 4.67T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 4.34T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 + 7.99T + 59T^{2} \) |
| 61 | \( 1 - 7.40T + 61T^{2} \) |
| 67 | \( 1 + 4.09T + 67T^{2} \) |
| 71 | \( 1 + 2.81T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 3.13T + 79T^{2} \) |
| 83 | \( 1 + 7.79T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
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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
−8.295142928318642613449193286102, −7.25549739181017920916711350280, −6.50578102960880527631764771978, −5.76051919958227618391145824893, −5.35419572283342321641460384786, −4.56442240256420616232242233002, −3.35706127958069607590970116606, −2.83553172569938472204272478977, −2.00494186119984524951402554669, −1.37611287938096751184673234872,
1.37611287938096751184673234872, 2.00494186119984524951402554669, 2.83553172569938472204272478977, 3.35706127958069607590970116606, 4.56442240256420616232242233002, 5.35419572283342321641460384786, 5.76051919958227618391145824893, 6.50578102960880527631764771978, 7.25549739181017920916711350280, 8.295142928318642613449193286102