| L(s) = 1 | + 27.9·2-s − 26.1·3-s + 270.·4-s + 368.·5-s − 731.·6-s − 6.74e3·8-s − 1.89e4·9-s + 1.02e4·10-s − 2.40e4·11-s − 7.08e3·12-s − 3.13e4·13-s − 9.62e3·15-s − 3.27e5·16-s + 5.55e5·17-s − 5.31e5·18-s − 8.37e5·19-s + 9.96e4·20-s − 6.72e5·22-s − 1.07e6·23-s + 1.76e5·24-s − 1.81e6·25-s − 8.75e5·26-s + 1.01e6·27-s − 2.75e6·29-s − 2.69e5·30-s − 5.81e6·31-s − 5.70e6·32-s + ⋯ |
| L(s) = 1 | + 1.23·2-s − 0.186·3-s + 0.528·4-s + 0.263·5-s − 0.230·6-s − 0.582·8-s − 0.965·9-s + 0.325·10-s − 0.494·11-s − 0.0986·12-s − 0.303·13-s − 0.0491·15-s − 1.24·16-s + 1.61·17-s − 1.19·18-s − 1.47·19-s + 0.139·20-s − 0.611·22-s − 0.797·23-s + 0.108·24-s − 0.930·25-s − 0.375·26-s + 0.366·27-s − 0.722·29-s − 0.0607·30-s − 1.13·31-s − 0.962·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 27.9T + 512T^{2} \) |
| 3 | \( 1 + 26.1T + 1.96e4T^{2} \) |
| 5 | \( 1 - 368.T + 1.95e6T^{2} \) |
| 11 | \( 1 + 2.40e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.13e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.55e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.37e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.07e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.75e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.81e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.20e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.79e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.48e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.63e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.12e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.65e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.20e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.11e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.60e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.69e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.44e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.05e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.38e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.25e8T + 7.60e17T^{2} \) |
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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
−13.07723702706783999849931803080, −12.21743818527279075886106215021, −11.05807809520340968437620313693, −9.540404864913532964552374728791, −7.987874219192515466451352304283, −6.10832942382151502367567975631, −5.35224445270165149579792986123, −3.83539173847634383419984069547, −2.42352235773818414390792106443, 0,
2.42352235773818414390792106443, 3.83539173847634383419984069547, 5.35224445270165149579792986123, 6.10832942382151502367567975631, 7.987874219192515466451352304283, 9.540404864913532964552374728791, 11.05807809520340968437620313693, 12.21743818527279075886106215021, 13.07723702706783999849931803080
