| L(s) = 1 | − 44.7·2-s + 202.·3-s + 1.48e3·4-s + 1.68e3·5-s − 9.06e3·6-s − 4.58e3·7-s − 4.37e4·8-s + 2.13e4·9-s − 7.54e4·10-s + 2.91e4·11-s + 3.01e5·12-s + 5.96e4·13-s + 2.05e5·14-s + 3.41e5·15-s + 1.19e6·16-s − 9.54e5·18-s + 3.93e5·19-s + 2.51e6·20-s − 9.29e5·21-s − 1.30e6·22-s − 5.81e5·23-s − 8.85e6·24-s + 8.87e5·25-s − 2.66e6·26-s + 3.33e5·27-s − 6.83e6·28-s + 5.69e6·29-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 1.44·3-s + 2.90·4-s + 1.20·5-s − 2.85·6-s − 0.722·7-s − 3.77·8-s + 1.08·9-s − 2.38·10-s + 0.600·11-s + 4.19·12-s + 0.579·13-s + 1.42·14-s + 1.74·15-s + 4.55·16-s − 2.14·18-s + 0.692·19-s + 3.50·20-s − 1.04·21-s − 1.18·22-s − 0.432·23-s − 5.44·24-s + 0.454·25-s − 1.14·26-s + 0.120·27-s − 2.10·28-s + 1.49·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.210620889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.210620889\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 44.7T + 512T^{2} \) |
| 3 | \( 1 - 202.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.68e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.58e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.91e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.96e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 3.93e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 5.81e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.69e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.57e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.31e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.59e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.10e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.90e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.95e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.61e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.62e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 7.28e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.46e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.69e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.32e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.39e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.69e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.49e6T + 7.60e17T^{2} \) |
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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
−9.702365403544383740900147062927, −9.367782078289547492453407658310, −8.598079443544614561886440232091, −7.78084151319366863446500284201, −6.68271455498329700077163252389, −5.98970239021509965711200943080, −3.40782804395879430873397479922, −2.58614224202588562155866981822, −1.76512590646713820946785290568, −0.849294514140058191149329469492,
0.849294514140058191149329469492, 1.76512590646713820946785290568, 2.58614224202588562155866981822, 3.40782804395879430873397479922, 5.98970239021509965711200943080, 6.68271455498329700077163252389, 7.78084151319366863446500284201, 8.598079443544614561886440232091, 9.367782078289547492453407658310, 9.702365403544383740900147062927
