| L(s) = 1 | + 4-s − 8·7-s − 3·9-s − 8·13-s + 16-s − 12·19-s + 6·25-s − 8·28-s − 3·36-s − 16·37-s − 16·43-s + 34·49-s − 8·52-s + 20·61-s + 24·63-s + 64-s + 12·67-s − 20·73-s − 12·76-s + 16·79-s + 9·81-s + 64·91-s − 2·97-s + 6·100-s − 20·109-s − 8·112-s + 24·117-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.02·7-s − 9-s − 2.21·13-s + 1/4·16-s − 2.75·19-s + 6/5·25-s − 1.51·28-s − 1/2·36-s − 2.63·37-s − 2.43·43-s + 34/7·49-s − 1.10·52-s + 2.56·61-s + 3.02·63-s + 1/8·64-s + 1.46·67-s − 2.34·73-s − 1.37·76-s + 1.80·79-s + 81-s + 6.70·91-s − 0.203·97-s + 3/5·100-s − 1.91·109-s − 0.755·112-s + 2.21·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 97 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.501039568706214050259380773636, −7.80295043300424113201528117600, −6.91525958798615443934798708079, −6.79502419315262861884017444820, −6.65007227051616693897794188818, −6.16068938165576146811569566993, −5.33519313821748327224317218590, −5.12634035972313084719024782667, −4.21007151259753375520969636989, −3.46279750097009321792818482789, −3.21113428811915609029373996367, −2.45314609137886810597617018889, −2.22991181864696470461280833479, 0, 0,
2.22991181864696470461280833479, 2.45314609137886810597617018889, 3.21113428811915609029373996367, 3.46279750097009321792818482789, 4.21007151259753375520969636989, 5.12634035972313084719024782667, 5.33519313821748327224317218590, 6.16068938165576146811569566993, 6.65007227051616693897794188818, 6.79502419315262861884017444820, 6.91525958798615443934798708079, 7.80295043300424113201528117600, 8.501039568706214050259380773636
