| L(s) = 1 | − 4-s − 5-s − 6·7-s − 4·9-s + 5·13-s − 3·16-s + 20-s − 3·23-s − 3·25-s + 6·28-s − 29-s + 6·35-s + 4·36-s + 4·45-s + 13·49-s − 5·52-s − 8·53-s − 8·59-s + 24·63-s + 7·64-s − 5·65-s + 11·67-s + 20·71-s + 3·80-s + 7·81-s + 18·83-s − 30·91-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.447·5-s − 2.26·7-s − 4/3·9-s + 1.38·13-s − 3/4·16-s + 0.223·20-s − 0.625·23-s − 3/5·25-s + 1.13·28-s − 0.185·29-s + 1.01·35-s + 2/3·36-s + 0.596·45-s + 13/7·49-s − 0.693·52-s − 1.09·53-s − 1.04·59-s + 3.02·63-s + 7/8·64-s − 0.620·65-s + 1.34·67-s + 2.37·71-s + 0.335·80-s + 7/9·81-s + 1.97·83-s − 3.14·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10933 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10933 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 132 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.00274500823544086006338237784, −10.82208890856886567537829541902, −9.802565269717364274116980350787, −9.432442717053843467843421263055, −9.080752251714741804081934892647, −8.290154120354716418454238759078, −7.951550605171018050800626255731, −6.77365830292056409944570659155, −6.39148144486217093494604677284, −5.93918958836021597292803536270, −5.08977600180795895939868749989, −3.82546822223363121729442740600, −3.57055238928682927694670032397, −2.63525639291551106854450960432, 0,
2.63525639291551106854450960432, 3.57055238928682927694670032397, 3.82546822223363121729442740600, 5.08977600180795895939868749989, 5.93918958836021597292803536270, 6.39148144486217093494604677284, 6.77365830292056409944570659155, 7.951550605171018050800626255731, 8.290154120354716418454238759078, 9.080752251714741804081934892647, 9.432442717053843467843421263055, 9.802565269717364274116980350787, 10.82208890856886567537829541902, 11.00274500823544086006338237784
