| L(s) = 1 | − 39.7·2-s + 81·3-s + 1.06e3·4-s − 1.87e3·5-s − 3.22e3·6-s − 43.1·7-s − 2.21e4·8-s + 6.56e3·9-s + 7.44e4·10-s + 7.61e4·11-s + 8.65e4·12-s − 4.37e4·13-s + 1.71e3·14-s − 1.51e5·15-s + 3.32e5·16-s + 4.20e5·17-s − 2.60e5·18-s − 3.33e5·19-s − 1.99e6·20-s − 3.49e3·21-s − 3.02e6·22-s − 1.14e5·23-s − 1.79e6·24-s + 1.54e6·25-s + 1.73e6·26-s + 5.31e5·27-s − 4.60e4·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 0.577·3-s + 2.08·4-s − 1.33·5-s − 1.01·6-s − 0.00678·7-s − 1.91·8-s + 0.333·9-s + 2.35·10-s + 1.56·11-s + 1.20·12-s − 0.424·13-s + 0.0119·14-s − 0.773·15-s + 1.26·16-s + 1.22·17-s − 0.585·18-s − 0.587·19-s − 2.79·20-s − 0.00391·21-s − 2.75·22-s − 0.0853·23-s − 1.10·24-s + 0.793·25-s + 0.746·26-s + 0.192·27-s − 0.0141·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.8494093527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8494093527\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 + 39.7T + 512T^{2} \) |
| 5 | \( 1 + 1.87e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 43.1T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.61e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 4.37e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.20e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.33e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.14e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.58e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.72e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.90e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.23e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.98e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.61e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.20e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 8.95e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.59e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.68e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.57e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.77e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.29e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.09e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.42e9T + 7.60e17T^{2} \) |
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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
−10.81275791545625444082402441068, −9.713545204518505802420440647856, −8.953210684785524302527711126009, −8.054434034755585921028361575639, −7.44091309717110451728852711432, −6.44150576637834649583393486641, −4.22970260743883712814296867937, −3.09468037411980301356373069132, −1.59831685085332918171660354573, −0.60328892023246022088976489070,
0.60328892023246022088976489070, 1.59831685085332918171660354573, 3.09468037411980301356373069132, 4.22970260743883712814296867937, 6.44150576637834649583393486641, 7.44091309717110451728852711432, 8.054434034755585921028361575639, 8.953210684785524302527711126009, 9.713545204518505802420440647856, 10.81275791545625444082402441068
