| L(s) = 1 | − 3.16·3-s + 0.368·5-s + 2.90·7-s + 7.01·9-s − 1.58·11-s + 13-s − 1.16·15-s + 6.69·17-s + 2.80·19-s − 9.18·21-s − 5.21·23-s − 4.86·25-s − 12.7·27-s + 7.21·29-s − 1.91·31-s + 5.01·33-s + 1.06·35-s − 10.3·37-s − 3.16·39-s − 2.48·41-s + 6.81·43-s + 2.58·45-s − 0.219·47-s + 1.41·49-s − 21.1·51-s + 6.48·53-s − 0.582·55-s + ⋯ |
| L(s) = 1 | − 1.82·3-s + 0.164·5-s + 1.09·7-s + 2.33·9-s − 0.477·11-s + 0.277·13-s − 0.300·15-s + 1.62·17-s + 0.642·19-s − 2.00·21-s − 1.08·23-s − 0.972·25-s − 2.44·27-s + 1.34·29-s − 0.343·31-s + 0.872·33-s + 0.180·35-s − 1.70·37-s − 0.506·39-s − 0.387·41-s + 1.03·43-s + 0.385·45-s − 0.0320·47-s + 0.201·49-s − 2.96·51-s + 0.890·53-s − 0.0785·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.085093366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085093366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 - 0.368T + 5T^{2} \) |
| 7 | \( 1 - 2.90T + 7T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 17 | \( 1 - 6.69T + 17T^{2} \) |
| 19 | \( 1 - 2.80T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 - 7.21T + 29T^{2} \) |
| 31 | \( 1 + 1.91T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 2.48T + 41T^{2} \) |
| 43 | \( 1 - 6.81T + 43T^{2} \) |
| 47 | \( 1 + 0.219T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 - 6.06T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 7.16T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 4.91T + 97T^{2} \) |
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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
−9.828193583929019773414863770722, −8.356674943397055349131046116608, −7.66953300417231027738670394298, −6.86754537772882592109656557687, −5.79264627345841825486312668838, −5.45179864831992569268159249074, −4.71210175664523741456186333065, −3.67196250477599130633792353459, −1.86709097128078622614143904284, −0.834205625393088810426308449046,
0.834205625393088810426308449046, 1.86709097128078622614143904284, 3.67196250477599130633792353459, 4.71210175664523741456186333065, 5.45179864831992569268159249074, 5.79264627345841825486312668838, 6.86754537772882592109656557687, 7.66953300417231027738670394298, 8.356674943397055349131046116608, 9.828193583929019773414863770722
