L(s) = 1 | + 4·5-s + 2·7-s − 6·9-s + 4·11-s − 16·19-s + 2·25-s + 8·35-s − 4·37-s + 8·43-s − 24·45-s + 3·49-s + 12·53-s + 16·55-s − 12·63-s + 8·77-s − 32·79-s + 27·81-s − 16·83-s − 12·89-s − 64·95-s − 12·97-s − 24·99-s + 24·107-s + 4·113-s + 5·121-s − 28·125-s + 127-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 0.755·7-s − 2·9-s + 1.20·11-s − 3.67·19-s + 2/5·25-s + 1.35·35-s − 0.657·37-s + 1.21·43-s − 3.57·45-s + 3/7·49-s + 1.64·53-s + 2.15·55-s − 1.51·63-s + 0.911·77-s − 3.60·79-s + 3·81-s − 1.75·83-s − 1.27·89-s − 6.56·95-s − 1.21·97-s − 2.41·99-s + 2.32·107-s + 0.376·113-s + 5/11·121-s − 2.50·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.423245666401170068214022844193, −7.69587064441186150768078543731, −6.88310002274215763839569601337, −6.67949194679031572918581545842, −5.94974241082854154889853475745, −5.82613210142107824134972732669, −5.72315905059004231786170242312, −4.86884684581528043387154729760, −4.18465932715971087267449515138, −4.03151585237256402081332864489, −2.98780927444278559643957768594, −2.30074942792092931708066021417, −2.14556791802447766709825303506, −1.44245425010921288807179765849, 0,
1.44245425010921288807179765849, 2.14556791802447766709825303506, 2.30074942792092931708066021417, 2.98780927444278559643957768594, 4.03151585237256402081332864489, 4.18465932715971087267449515138, 4.86884684581528043387154729760, 5.72315905059004231786170242312, 5.82613210142107824134972732669, 5.94974241082854154889853475745, 6.67949194679031572918581545842, 6.88310002274215763839569601337, 7.69587064441186150768078543731, 8.423245666401170068214022844193