| L(s) = 1 | − 2.41·2-s − 2.17·4-s + 19.8·5-s + 24.5·8-s − 48.0·10-s − 23.9·11-s − 87.3·13-s − 41.9·16-s − 5.63·17-s − 64.8·19-s − 43.2·20-s + 57.7·22-s + 25.5·23-s + 270.·25-s + 210.·26-s − 60.3·29-s + 122.·31-s − 95.2·32-s + 13.6·34-s − 56.1·37-s + 156.·38-s + 488.·40-s − 299.·41-s − 501.·43-s + 51.9·44-s − 61.7·46-s + 305.·47-s + ⋯ |
| L(s) = 1 | − 0.853·2-s − 0.271·4-s + 1.77·5-s + 1.08·8-s − 1.51·10-s − 0.656·11-s − 1.86·13-s − 0.654·16-s − 0.0804·17-s − 0.783·19-s − 0.483·20-s + 0.560·22-s + 0.232·23-s + 2.16·25-s + 1.59·26-s − 0.386·29-s + 0.710·31-s − 0.526·32-s + 0.0686·34-s − 0.249·37-s + 0.668·38-s + 1.93·40-s − 1.14·41-s − 1.77·43-s + 0.178·44-s − 0.198·46-s + 0.948·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.41T + 8T^{2} \) |
| 5 | \( 1 - 19.8T + 125T^{2} \) |
| 11 | \( 1 + 23.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 87.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.63T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 25.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 60.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 56.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 501.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 375.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 627.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 3.75T + 2.26e5T^{2} \) |
| 67 | \( 1 + 813.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 165.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 619.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 138.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 621.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 285.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 603.T + 9.12e5T^{2} \) |
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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
−10.25115140680928069793907822682, −9.424361388081104139528733568921, −8.724496899951372790870647962732, −7.59559597392885505268808199278, −6.62161073278584738876393138224, −5.36998073281495201553786291057, −4.71367889827278896047239244691, −2.62248657262314888058741630356, −1.67612468822529134862330873373, 0,
1.67612468822529134862330873373, 2.62248657262314888058741630356, 4.71367889827278896047239244691, 5.36998073281495201553786291057, 6.62161073278584738876393138224, 7.59559597392885505268808199278, 8.724496899951372790870647962732, 9.424361388081104139528733568921, 10.25115140680928069793907822682
