| L(s) = 1 | − 10·2-s + 27·3-s − 28·4-s − 410·5-s − 270·6-s + 1.02e3·7-s + 1.56e3·8-s + 729·9-s + 4.10e3·10-s − 756·12-s − 1.29e4·13-s − 1.02e4·14-s − 1.10e4·15-s − 1.20e4·16-s − 1.70e4·17-s − 7.29e3·18-s + 5.41e4·19-s + 1.14e4·20-s + 2.77e4·21-s − 1.14e4·23-s + 4.21e4·24-s + 8.99e4·25-s + 1.29e5·26-s + 1.96e4·27-s − 2.87e4·28-s + 1.86e5·29-s + 1.10e5·30-s + ⋯ |
| L(s) = 1 | − 0.883·2-s + 0.577·3-s − 0.218·4-s − 1.46·5-s − 0.510·6-s + 1.13·7-s + 1.07·8-s + 1/3·9-s + 1.29·10-s − 0.126·12-s − 1.63·13-s − 1.00·14-s − 0.846·15-s − 0.733·16-s − 0.842·17-s − 0.294·18-s + 1.81·19-s + 0.320·20-s + 0.654·21-s − 0.196·23-s + 0.621·24-s + 1.15·25-s + 1.44·26-s + 0.192·27-s − 0.247·28-s + 1.42·29-s + 0.748·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8400009340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8400009340\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{3} T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 5 p T + p^{7} T^{2} \) |
| 5 | \( 1 + 82 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 1028 T + p^{7} T^{2} \) |
| 13 | \( 1 + 12958 T + p^{7} T^{2} \) |
| 17 | \( 1 + 17062 T + p^{7} T^{2} \) |
| 19 | \( 1 - 54168 T + p^{7} T^{2} \) |
| 23 | \( 1 + 11488 T + p^{7} T^{2} \) |
| 29 | \( 1 - 186654 T + p^{7} T^{2} \) |
| 31 | \( 1 + 188672 T + p^{7} T^{2} \) |
| 37 | \( 1 - 395886 T + p^{7} T^{2} \) |
| 41 | \( 1 - 47546 T + p^{7} T^{2} \) |
| 43 | \( 1 + 602088 T + p^{7} T^{2} \) |
| 47 | \( 1 + 647200 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1312722 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2681140 T + p^{7} T^{2} \) |
| 61 | \( 1 + 551190 T + p^{7} T^{2} \) |
| 67 | \( 1 - 459260 T + p^{7} T^{2} \) |
| 71 | \( 1 + 18072 T + p^{7} T^{2} \) |
| 73 | \( 1 - 426062 T + p^{7} T^{2} \) |
| 79 | \( 1 + 297764 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5684028 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6342966 T + p^{7} T^{2} \) |
| 97 | \( 1 - 16651586 T + p^{7} T^{2} \) |
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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
−9.997884280592534766089950900270, −9.206113913547255090649871675060, −8.200148730076443853886928082911, −7.74892217963738285862215137078, −7.15371358570806111149218551733, −4.87648391738558157832800548006, −4.49346523603011971445142095832, −3.12156903138646028739567282852, −1.69020859825690454856093285298, −0.49214082847864347035526580154,
0.49214082847864347035526580154, 1.69020859825690454856093285298, 3.12156903138646028739567282852, 4.49346523603011971445142095832, 4.87648391738558157832800548006, 7.15371358570806111149218551733, 7.74892217963738285862215137078, 8.200148730076443853886928082911, 9.206113913547255090649871675060, 9.997884280592534766089950900270
