| L(s) = 1 | − 2·2-s + 3·4-s + 7-s − 4·8-s − 2·9-s − 2·14-s + 5·16-s + 12·17-s + 4·18-s + 4·19-s − 10·25-s + 3·28-s − 12·29-s − 6·32-s − 24·34-s − 6·36-s − 8·38-s + 49-s + 20·50-s − 4·56-s + 24·58-s + 16·61-s − 2·63-s + 7·64-s + 36·68-s + 8·72-s + 12·76-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s − 2/3·9-s − 0.534·14-s + 5/4·16-s + 2.91·17-s + 0.942·18-s + 0.917·19-s − 2·25-s + 0.566·28-s − 2.22·29-s − 1.06·32-s − 4.11·34-s − 36-s − 1.29·38-s + 1/7·49-s + 2.82·50-s − 0.534·56-s + 3.15·58-s + 2.04·61-s − 0.251·63-s + 7/8·64-s + 4.36·68-s + 0.942·72-s + 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| 23 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.110149332059267353023137382824, −7.59769760654238825164816496129, −7.57571100088867902110310233811, −7.01062402394523129137998185488, −6.33381612131422841357801936519, −5.57928681742950427486583645839, −5.57132308384623995366860612457, −5.32616093306835891028088920065, −3.97448209386129731451102084779, −3.73919187382497721540937213023, −3.06849498900201108772759649392, −2.49577833291038984931506482388, −1.63743099327135487468573944042, −1.18173816534949059288556508761, 0,
1.18173816534949059288556508761, 1.63743099327135487468573944042, 2.49577833291038984931506482388, 3.06849498900201108772759649392, 3.73919187382497721540937213023, 3.97448209386129731451102084779, 5.32616093306835891028088920065, 5.57132308384623995366860612457, 5.57928681742950427486583645839, 6.33381612131422841357801936519, 7.01062402394523129137998185488, 7.57571100088867902110310233811, 7.59769760654238825164816496129, 8.110149332059267353023137382824
