| L(s) = 1 | + 2·3-s + 4-s + 2·5-s + 3·9-s − 3·11-s + 2·12-s + 4·15-s + 16-s + 2·20-s + 8·23-s − 7·25-s + 4·27-s + 2·31-s − 6·33-s + 3·36-s − 4·37-s − 3·44-s + 6·45-s − 14·47-s + 2·48-s − 10·49-s + 28·53-s − 6·55-s + 20·59-s + 4·60-s + 64-s − 14·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 0.894·5-s + 9-s − 0.904·11-s + 0.577·12-s + 1.03·15-s + 1/4·16-s + 0.447·20-s + 1.66·23-s − 7/5·25-s + 0.769·27-s + 0.359·31-s − 1.04·33-s + 1/2·36-s − 0.657·37-s − 0.452·44-s + 0.894·45-s − 2.04·47-s + 0.288·48-s − 1.42·49-s + 3.84·53-s − 0.809·55-s + 2.60·59-s + 0.516·60-s + 1/8·64-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4186116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4186116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.261387361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.261387361\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−7.34984076411643108613521427718, −7.33319367325295365011876797445, −6.54724301852172258049775142993, −6.21449897841157477524696518952, −5.85900673940507999685378798678, −5.17708288135546094293333366874, −5.03620516471027669405867234682, −4.51503419048647972746572531845, −3.64970112975061735540208218245, −3.54463186962642716865631176024, −2.94295235845990683987012556381, −2.34546937125578111204655036909, −2.15807062856181143969411435609, −1.57905538969586167062740319602, −0.74067976683628195306026315515,
0.74067976683628195306026315515, 1.57905538969586167062740319602, 2.15807062856181143969411435609, 2.34546937125578111204655036909, 2.94295235845990683987012556381, 3.54463186962642716865631176024, 3.64970112975061735540208218245, 4.51503419048647972746572531845, 5.03620516471027669405867234682, 5.17708288135546094293333366874, 5.85900673940507999685378798678, 6.21449897841157477524696518952, 6.54724301852172258049775142993, 7.33319367325295365011876797445, 7.34984076411643108613521427718
