| L(s) = 1 | − 5·9-s + 2·13-s − 10·17-s − 10·25-s − 6·29-s + 4·37-s − 16·41-s − 5·49-s + 18·53-s + 28·61-s + 18·73-s + 16·81-s − 24·89-s + 28·97-s − 28·101-s + 14·109-s − 36·113-s − 10·117-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 50·153-s + 157-s + ⋯ |
| L(s) = 1 | − 5/3·9-s + 0.554·13-s − 2.42·17-s − 2·25-s − 1.11·29-s + 0.657·37-s − 2.49·41-s − 5/7·49-s + 2.47·53-s + 3.58·61-s + 2.10·73-s + 16/9·81-s − 2.54·89-s + 2.84·97-s − 2.78·101-s + 1.34·109-s − 3.38·113-s − 0.924·117-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 4.04·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.815161388517683585055924650409, −9.389846141086456194104312076880, −8.649214735442593578398627320691, −8.517458100677749789097011376696, −8.058703950373762851562620749027, −7.18297506166153508271072468025, −6.61570751747134755245908356006, −6.21150502900434643631796858377, −5.41676012615338774687665516924, −5.18915175026014233079805145903, −3.93346432196685798564362812764, −3.80299080464445309990426195305, −2.56005012019129152265351485884, −2.06113805202725322767384962385, 0,
2.06113805202725322767384962385, 2.56005012019129152265351485884, 3.80299080464445309990426195305, 3.93346432196685798564362812764, 5.18915175026014233079805145903, 5.41676012615338774687665516924, 6.21150502900434643631796858377, 6.61570751747134755245908356006, 7.18297506166153508271072468025, 8.058703950373762851562620749027, 8.517458100677749789097011376696, 8.649214735442593578398627320691, 9.389846141086456194104312076880, 9.815161388517683585055924650409
