| L(s) = 1 | − 2·3-s − 3·4-s + 5-s + 3·9-s + 6·12-s − 2·15-s + 5·16-s + 2·17-s + 8·19-s − 3·20-s + 25-s − 4·27-s − 9·36-s − 20·37-s + 3·45-s − 10·48-s − 14·49-s − 4·51-s − 16·57-s − 8·59-s + 6·60-s − 3·64-s − 6·68-s + 20·73-s − 2·75-s − 24·76-s + 5·80-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3/2·4-s + 0.447·5-s + 9-s + 1.73·12-s − 0.516·15-s + 5/4·16-s + 0.485·17-s + 1.83·19-s − 0.670·20-s + 1/5·25-s − 0.769·27-s − 3/2·36-s − 3.28·37-s + 0.447·45-s − 1.44·48-s − 2·49-s − 0.560·51-s − 2.11·57-s − 1.04·59-s + 0.774·60-s − 3/8·64-s − 0.727·68-s + 2.34·73-s − 0.230·75-s − 2.75·76-s + 0.559·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.609806086799085606026396632541, −8.104875420158931912775695831681, −7.66488013441745380243230842523, −7.02706467814418307736798527147, −6.66671811702300585611926997110, −6.03675543912362505739101610093, −5.36202152889991118989599873720, −5.23920392624592057055772361749, −4.91593057103779961454763865656, −4.22289405590317380357585208255, −3.50942513045001276890420119823, −3.18074927530238362991828410929, −1.80969389884009513212184610049, −1.08138444197682596325470232778, 0,
1.08138444197682596325470232778, 1.80969389884009513212184610049, 3.18074927530238362991828410929, 3.50942513045001276890420119823, 4.22289405590317380357585208255, 4.91593057103779961454763865656, 5.23920392624592057055772361749, 5.36202152889991118989599873720, 6.03675543912362505739101610093, 6.66671811702300585611926997110, 7.02706467814418307736798527147, 7.66488013441745380243230842523, 8.104875420158931912775695831681, 8.609806086799085606026396632541
