L(s) = 1 | + 1.87·3-s + 2.83·5-s + 3.83·7-s + 0.498·9-s + 11-s + 6.12·13-s + 5.29·15-s − 3.70·17-s − 19-s + 7.16·21-s − 1.01·23-s + 3.02·25-s − 4.67·27-s − 3.56·29-s + 5.29·31-s + 1.87·33-s + 10.8·35-s + 1.68·37-s + 11.4·39-s − 6.71·41-s + 3.59·43-s + 1.41·45-s + 6.48·47-s + 7.69·49-s − 6.93·51-s − 6.34·53-s + 2.83·55-s + ⋯ |
L(s) = 1 | + 1.07·3-s + 1.26·5-s + 1.44·7-s + 0.166·9-s + 0.301·11-s + 1.69·13-s + 1.36·15-s − 0.899·17-s − 0.229·19-s + 1.56·21-s − 0.212·23-s + 0.605·25-s − 0.900·27-s − 0.662·29-s + 0.951·31-s + 0.325·33-s + 1.83·35-s + 0.277·37-s + 1.83·39-s − 1.04·41-s + 0.548·43-s + 0.210·45-s + 0.945·47-s + 1.09·49-s − 0.971·51-s − 0.872·53-s + 0.382·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.334723356\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.334723356\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 1.87T + 3T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 13 | \( 1 - 6.12T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 - 1.68T + 37T^{2} \) |
| 41 | \( 1 + 6.71T + 41T^{2} \) |
| 43 | \( 1 - 3.59T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 6.34T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 0.743T + 61T^{2} \) |
| 67 | \( 1 + 2.07T + 67T^{2} \) |
| 71 | \( 1 - 2.65T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 1.81T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 1.51T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−8.566586289116982170861515972938, −8.226674235382473632492461053914, −7.27318766235173818187019682962, −6.19581209991098071507790604596, −5.76692882760851629555477482290, −4.68909437009930878556763349813, −3.92719954471507299255435539368, −2.85711174793307199484108071881, −1.92957895989215285813473565454, −1.43190378759237055537892297456,
1.43190378759237055537892297456, 1.92957895989215285813473565454, 2.85711174793307199484108071881, 3.92719954471507299255435539368, 4.68909437009930878556763349813, 5.76692882760851629555477482290, 6.19581209991098071507790604596, 7.27318766235173818187019682962, 8.226674235382473632492461053914, 8.566586289116982170861515972938