| L(s) = 1 | − 3-s + 5-s + 1.32·7-s + 9-s − 0.398·11-s + 2.92·13-s − 15-s + 17-s + 7.97·19-s − 1.32·21-s + 5.72·23-s + 25-s − 27-s − 1.47·29-s − 7.57·31-s + 0.398·33-s + 1.32·35-s − 1.04·37-s − 2.92·39-s + 5.97·41-s − 7.44·43-s + 45-s + 4.39·47-s − 5.24·49-s − 51-s − 0.676·53-s − 0.398·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.500·7-s + 0.333·9-s − 0.120·11-s + 0.811·13-s − 0.258·15-s + 0.242·17-s + 1.82·19-s − 0.288·21-s + 1.19·23-s + 0.200·25-s − 0.192·27-s − 0.273·29-s − 1.35·31-s + 0.0693·33-s + 0.223·35-s − 0.171·37-s − 0.468·39-s + 0.932·41-s − 1.13·43-s + 0.149·45-s + 0.641·47-s − 0.749·49-s − 0.140·51-s − 0.0929·53-s − 0.0536·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.071784727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.071784727\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 + 0.398T + 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 19 | \( 1 - 7.97T + 19T^{2} \) |
| 23 | \( 1 - 5.72T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 + 7.57T + 31T^{2} \) |
| 37 | \( 1 + 1.04T + 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 43 | \( 1 + 7.44T + 43T^{2} \) |
| 47 | \( 1 - 4.39T + 47T^{2} \) |
| 53 | \( 1 + 0.676T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 4.51T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 1.85T + 71T^{2} \) |
| 73 | \( 1 - 7.19T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 - 7.44T + 83T^{2} \) |
| 89 | \( 1 + 3.72T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−8.460407106852897302671931503813, −7.53782560405908655124621674902, −7.04410542379849853305293466357, −6.08791758461656880761209990787, −5.39185618883855377195231876784, −4.97224461028622060031242093244, −3.81750555111990832735102918117, −3.01507397082993509150361007057, −1.73598228317017315055499854873, −0.912569099025004708236312435200,
0.912569099025004708236312435200, 1.73598228317017315055499854873, 3.01507397082993509150361007057, 3.81750555111990832735102918117, 4.97224461028622060031242093244, 5.39185618883855377195231876784, 6.08791758461656880761209990787, 7.04410542379849853305293466357, 7.53782560405908655124621674902, 8.460407106852897302671931503813
