| L(s) = 1 | + 0.452·2-s + 1.76·3-s − 1.79·4-s + 4.27·5-s + 0.800·6-s − 1.71·8-s + 0.125·9-s + 1.93·10-s − 3.08·11-s − 3.17·12-s + 0.931·13-s + 7.55·15-s + 2.81·16-s + 3.79·17-s + 0.0568·18-s − 7.95·19-s − 7.67·20-s − 1.39·22-s + 8.86·23-s − 3.03·24-s + 13.2·25-s + 0.421·26-s − 5.08·27-s − 3.21·29-s + 3.42·30-s + 7.47·31-s + 4.70·32-s + ⋯ |
| L(s) = 1 | + 0.319·2-s + 1.02·3-s − 0.897·4-s + 1.91·5-s + 0.326·6-s − 0.607·8-s + 0.0418·9-s + 0.611·10-s − 0.929·11-s − 0.916·12-s + 0.258·13-s + 1.95·15-s + 0.703·16-s + 0.919·17-s + 0.0134·18-s − 1.82·19-s − 1.71·20-s − 0.297·22-s + 1.84·23-s − 0.619·24-s + 2.65·25-s + 0.0826·26-s − 0.977·27-s − 0.597·29-s + 0.624·30-s + 1.34·31-s + 0.832·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.524545679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.524545679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 0.452T + 2T^{2} \) |
| 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 - 4.27T + 5T^{2} \) |
| 11 | \( 1 + 3.08T + 11T^{2} \) |
| 13 | \( 1 - 0.931T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 + 7.95T + 19T^{2} \) |
| 23 | \( 1 - 8.86T + 23T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 + 0.130T + 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 - 8.32T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 0.606T + 53T^{2} \) |
| 59 | \( 1 - 7.07T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 79 | \( 1 + 4.88T + 79T^{2} \) |
| 83 | \( 1 + 6.49T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 1.38T + 97T^{2} \) |
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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
−8.708184640515453830636361953428, −8.145699841767150534947238755422, −7.04910998424279091522765665172, −6.02845675763267311126487276837, −5.56444746042293811081340863499, −4.86066384538048673119891676387, −3.84366364936654894749431185443, −2.71475256109750892046744208857, −2.43747626075444410142564045042, −1.05122933071833569080307933781,
1.05122933071833569080307933781, 2.43747626075444410142564045042, 2.71475256109750892046744208857, 3.84366364936654894749431185443, 4.86066384538048673119891676387, 5.56444746042293811081340863499, 6.02845675763267311126487276837, 7.04910998424279091522765665172, 8.145699841767150534947238755422, 8.708184640515453830636361953428
