| L(s) = 1 | − 32·2-s − 12·3-s + 1.02e3·4-s + 3.12e3·5-s + 384·6-s − 1.41e4·7-s − 3.27e4·8-s − 1.77e5·9-s − 1.00e5·10-s − 7.56e5·11-s − 1.22e4·12-s − 9.05e5·13-s + 4.53e5·14-s − 3.75e4·15-s + 1.04e6·16-s + 2.80e6·17-s + 5.66e6·18-s − 5.42e6·19-s + 3.20e6·20-s + 1.70e5·21-s + 2.42e7·22-s − 1.02e7·23-s + 3.93e5·24-s + 9.76e6·25-s + 2.89e7·26-s + 4.24e6·27-s − 1.45e7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0285·3-s + 1/2·4-s + 0.447·5-s + 0.0201·6-s − 0.318·7-s − 0.353·8-s − 0.999·9-s − 0.316·10-s − 1.41·11-s − 0.0142·12-s − 0.676·13-s + 0.225·14-s − 0.0127·15-s + 1/4·16-s + 0.478·17-s + 0.706·18-s − 0.502·19-s + 0.223·20-s + 0.00908·21-s + 1.00·22-s − 0.331·23-s + 0.0100·24-s + 1/5·25-s + 0.478·26-s + 0.0569·27-s − 0.159·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p^{5} T \) |
| 5 | \( 1 - p^{5} T \) |
| good | 3 | \( 1 + 4 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 14176 T + p^{11} T^{2} \) |
| 11 | \( 1 + 756348 T + p^{11} T^{2} \) |
| 13 | \( 1 + 905482 T + p^{11} T^{2} \) |
| 17 | \( 1 - 2803794 T + p^{11} T^{2} \) |
| 19 | \( 1 + 5428660 T + p^{11} T^{2} \) |
| 23 | \( 1 + 10236672 T + p^{11} T^{2} \) |
| 29 | \( 1 + 197498010 T + p^{11} T^{2} \) |
| 31 | \( 1 + 44362288 T + p^{11} T^{2} \) |
| 37 | \( 1 - 576737054 T + p^{11} T^{2} \) |
| 41 | \( 1 - 930058362 T + p^{11} T^{2} \) |
| 43 | \( 1 - 1605598988 T + p^{11} T^{2} \) |
| 47 | \( 1 + 1803684456 T + p^{11} T^{2} \) |
| 53 | \( 1 - 1558674798 T + p^{11} T^{2} \) |
| 59 | \( 1 + 9501997020 T + p^{11} T^{2} \) |
| 61 | \( 1 - 6736320422 T + p^{11} T^{2} \) |
| 67 | \( 1 - 8402906564 T + p^{11} T^{2} \) |
| 71 | \( 1 + 4806306168 T + p^{11} T^{2} \) |
| 73 | \( 1 - 7462713338 T + p^{11} T^{2} \) |
| 79 | \( 1 + 20644540720 T + p^{11} T^{2} \) |
| 83 | \( 1 + 68013349212 T + p^{11} T^{2} \) |
| 89 | \( 1 - 69871323210 T + p^{11} T^{2} \) |
| 97 | \( 1 - 39960952514 T + p^{11} T^{2} \) |
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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
−17.48930768525424838624974260424, −16.30903184694101713239260185371, −14.66304898227211217643599468800, −12.84935111026530838449317072989, −10.99119727523260327705273438937, −9.534132912224645339292975549663, −7.81357258564763348519806998105, −5.72204230345625588446215424836, −2.54864493510662630114409978563, 0,
2.54864493510662630114409978563, 5.72204230345625588446215424836, 7.81357258564763348519806998105, 9.534132912224645339292975549663, 10.99119727523260327705273438937, 12.84935111026530838449317072989, 14.66304898227211217643599468800, 16.30903184694101713239260185371, 17.48930768525424838624974260424
