| L(s) = 1 | + 10·11-s − 4·16-s + 10·19-s + 10·29-s − 18·31-s + 14·41-s + 14·49-s + 2·59-s − 4·61-s + 2·71-s − 24·79-s + 18·89-s − 6·101-s + 2·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s − 40·176-s + ⋯ |
| L(s) = 1 | + 3.01·11-s − 16-s + 2.29·19-s + 1.85·29-s − 3.23·31-s + 2.18·41-s + 2·49-s + 0.260·59-s − 0.512·61-s + 0.237·71-s − 2.70·79-s + 1.90·89-s − 0.597·101-s + 0.191·109-s + 4.81·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s − 3.01·176-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.572526578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.572526578\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.334098062684827383259916935914, −9.052150047314487167564381113354, −8.647564499423975792781250859877, −8.416621609205187749505237078390, −7.41230652305881932370041170001, −7.39113424670458149948219062461, −7.10461552715674291774889312507, −6.68591512574905174627733785393, −6.13681259531888009002934379624, −5.77776791828149324925887121240, −5.53869196918469484730253253699, −4.69640535634946212378317671111, −4.50504761785847529968900571770, −3.85317928540879815311252466217, −3.70273475350232943408782369464, −3.13986918992379447472209950127, −2.49273549885662812806547616181, −1.80871195297257303763879179492, −1.22901712178924335094102592129, −0.793629708182262923546600713212,
0.793629708182262923546600713212, 1.22901712178924335094102592129, 1.80871195297257303763879179492, 2.49273549885662812806547616181, 3.13986918992379447472209950127, 3.70273475350232943408782369464, 3.85317928540879815311252466217, 4.50504761785847529968900571770, 4.69640535634946212378317671111, 5.53869196918469484730253253699, 5.77776791828149324925887121240, 6.13681259531888009002934379624, 6.68591512574905174627733785393, 7.10461552715674291774889312507, 7.39113424670458149948219062461, 7.41230652305881932370041170001, 8.416621609205187749505237078390, 8.647564499423975792781250859877, 9.052150047314487167564381113354, 9.334098062684827383259916935914
