| L(s) = 1 | − 56·2-s − 243·3-s + 1.08e3·4-s + 3.12e3·5-s + 1.36e4·6-s + 2.79e4·7-s + 5.37e4·8-s + 5.90e4·9-s − 1.75e5·10-s − 1.12e5·11-s − 2.64e5·12-s − 1.09e6·13-s − 1.56e6·14-s − 7.59e5·15-s − 5.23e6·16-s − 2.49e5·17-s − 3.30e6·18-s − 1.37e7·19-s + 3.40e6·20-s − 6.80e6·21-s + 6.27e6·22-s + 4.13e7·23-s − 1.30e7·24-s + 9.76e6·25-s + 6.14e7·26-s − 1.43e7·27-s + 3.04e7·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 0.577·3-s + 0.531·4-s + 0.447·5-s + 0.714·6-s + 0.629·7-s + 0.580·8-s + 1/3·9-s − 0.553·10-s − 0.209·11-s − 0.306·12-s − 0.819·13-s − 0.778·14-s − 0.258·15-s − 1.24·16-s − 0.0426·17-s − 0.412·18-s − 1.27·19-s + 0.237·20-s − 0.363·21-s + 0.259·22-s + 1.34·23-s − 0.334·24-s + 1/5·25-s + 1.01·26-s − 0.192·27-s + 0.334·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{5} T \) |
| 5 | \( 1 - p^{5} T \) |
| good | 2 | \( 1 + 7 p^{3} T + p^{11} T^{2} \) |
| 7 | \( 1 - 27984 T + p^{11} T^{2} \) |
| 11 | \( 1 + 112028 T + p^{11} T^{2} \) |
| 13 | \( 1 + 1096922 T + p^{11} T^{2} \) |
| 17 | \( 1 + 249566 T + p^{11} T^{2} \) |
| 19 | \( 1 + 13712420 T + p^{11} T^{2} \) |
| 23 | \( 1 - 41395728 T + p^{11} T^{2} \) |
| 29 | \( 1 + 4533850 T + p^{11} T^{2} \) |
| 31 | \( 1 + 265339008 T + p^{11} T^{2} \) |
| 37 | \( 1 + 212136946 T + p^{11} T^{2} \) |
| 41 | \( 1 + 1266969958 T + p^{11} T^{2} \) |
| 43 | \( 1 - 14129548 T + p^{11} T^{2} \) |
| 47 | \( 1 + 2657273336 T + p^{11} T^{2} \) |
| 53 | \( 1 - 2402699278 T + p^{11} T^{2} \) |
| 59 | \( 1 - 7498737220 T + p^{11} T^{2} \) |
| 61 | \( 1 + 4064828858 T + p^{11} T^{2} \) |
| 67 | \( 1 - 6871514244 T + p^{11} T^{2} \) |
| 71 | \( 1 + 13283734648 T + p^{11} T^{2} \) |
| 73 | \( 1 + 28875844262 T + p^{11} T^{2} \) |
| 79 | \( 1 - 27100302240 T + p^{11} T^{2} \) |
| 83 | \( 1 + 34365255132 T + p^{11} T^{2} \) |
| 89 | \( 1 + 63500412630 T + p^{11} T^{2} \) |
| 97 | \( 1 - 19634495234 T + p^{11} T^{2} \) |
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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
−16.63182495259775246427154168707, −14.82166084129453650647384718757, −13.02286657349237733147212992755, −11.19569489652602879008610450418, −10.08917067549954039747178841948, −8.659063713795335150329635271913, −7.07544833509577549079444966194, −4.96727937920632166045052817687, −1.74079702699032931824273264204, 0,
1.74079702699032931824273264204, 4.96727937920632166045052817687, 7.07544833509577549079444966194, 8.659063713795335150329635271913, 10.08917067549954039747178841948, 11.19569489652602879008610450418, 13.02286657349237733147212992755, 14.82166084129453650647384718757, 16.63182495259775246427154168707
