| L(s) = 1 | + 2-s + 2.37·3-s + 4-s + 4.07·5-s + 2.37·6-s − 1.94·7-s + 8-s + 2.64·9-s + 4.07·10-s − 2.44·11-s + 2.37·12-s + 1.35·13-s − 1.94·14-s + 9.67·15-s + 16-s − 5.09·17-s + 2.64·18-s − 3.55·19-s + 4.07·20-s − 4.62·21-s − 2.44·22-s + 2.37·24-s + 11.5·25-s + 1.35·26-s − 0.846·27-s − 1.94·28-s − 4.40·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.37·3-s + 0.5·4-s + 1.82·5-s + 0.969·6-s − 0.735·7-s + 0.353·8-s + 0.881·9-s + 1.28·10-s − 0.738·11-s + 0.685·12-s + 0.376·13-s − 0.519·14-s + 2.49·15-s + 0.250·16-s − 1.23·17-s + 0.623·18-s − 0.815·19-s + 0.910·20-s − 1.00·21-s − 0.522·22-s + 0.484·24-s + 2.31·25-s + 0.266·26-s − 0.162·27-s − 0.367·28-s − 0.817·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.490524645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.490524645\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.37T + 3T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 7 | \( 1 + 1.94T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 29 | \( 1 + 4.40T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 1.94T + 37T^{2} \) |
| 41 | \( 1 + 5.57T + 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 + 2.29T + 47T^{2} \) |
| 53 | \( 1 - 8.84T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 5.96T + 61T^{2} \) |
| 67 | \( 1 + 8.83T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 4.92T + 73T^{2} \) |
| 79 | \( 1 + 7.06T + 79T^{2} \) |
| 83 | \( 1 + 4.96T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 8.59T + 97T^{2} \) |
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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
−9.986716168501808741683154296041, −8.969702442961656440461244141049, −8.560107715359345216361405801634, −7.23698678145384824922411227322, −6.39121347154709170576232238340, −5.72561776927678829548178726215, −4.61236107001189228301521193477, −3.38831153216863246886020998700, −2.49484509572493128102453217616, −1.94415903866258520207210824530,
1.94415903866258520207210824530, 2.49484509572493128102453217616, 3.38831153216863246886020998700, 4.61236107001189228301521193477, 5.72561776927678829548178726215, 6.39121347154709170576232238340, 7.23698678145384824922411227322, 8.560107715359345216361405801634, 8.969702442961656440461244141049, 9.986716168501808741683154296041
