| L(s) = 1 | + 0.352·2-s − 1.11·3-s − 1.87·4-s + 5-s − 0.392·6-s + 7-s − 1.36·8-s − 1.76·9-s + 0.352·10-s + 2.08·12-s − 1.18·13-s + 0.352·14-s − 1.11·15-s + 3.26·16-s + 2.58·17-s − 0.621·18-s − 2.84·19-s − 1.87·20-s − 1.11·21-s − 2.43·23-s + 1.52·24-s + 25-s − 0.418·26-s + 5.30·27-s − 1.87·28-s − 0.595·29-s − 0.392·30-s + ⋯ |
| L(s) = 1 | + 0.249·2-s − 0.642·3-s − 0.937·4-s + 0.447·5-s − 0.160·6-s + 0.377·7-s − 0.483·8-s − 0.586·9-s + 0.111·10-s + 0.602·12-s − 0.329·13-s + 0.0943·14-s − 0.287·15-s + 0.817·16-s + 0.625·17-s − 0.146·18-s − 0.653·19-s − 0.419·20-s − 0.242·21-s − 0.508·23-s + 0.310·24-s + 0.200·25-s − 0.0821·26-s + 1.01·27-s − 0.354·28-s − 0.110·29-s − 0.0717·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.022038212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.022038212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.352T + 2T^{2} \) |
| 3 | \( 1 + 1.11T + 3T^{2} \) |
| 13 | \( 1 + 1.18T + 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 + 2.84T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 0.595T + 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 + 1.54T + 37T^{2} \) |
| 41 | \( 1 + 1.81T + 41T^{2} \) |
| 43 | \( 1 + 5.14T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 + 3.64T + 61T^{2} \) |
| 67 | \( 1 + 0.843T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 4.54T + 73T^{2} \) |
| 79 | \( 1 - 0.402T + 79T^{2} \) |
| 83 | \( 1 + 0.473T + 83T^{2} \) |
| 89 | \( 1 + 1.45T + 89T^{2} \) |
| 97 | \( 1 - 6.40T + 97T^{2} \) |
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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
−8.464653972908514694608016091539, −7.77307301423722402900908391696, −6.77341391902308531180710061070, −5.89316022384238979573365345962, −5.45530063988146312303019643355, −4.81892333971977687673386607648, −3.96930531350578246814510741089, −3.05396604416467972765388438726, −1.89375838049902819671880986165, −0.56599631142840088546115966820,
0.56599631142840088546115966820, 1.89375838049902819671880986165, 3.05396604416467972765388438726, 3.96930531350578246814510741089, 4.81892333971977687673386607648, 5.45530063988146312303019643355, 5.89316022384238979573365345962, 6.77341391902308531180710061070, 7.77307301423722402900908391696, 8.464653972908514694608016091539
