| L(s) = 1 | + 4·4-s + 3·5-s − 5·9-s + 12·16-s + 12·20-s + 4·25-s + 10·31-s − 20·36-s − 15·45-s + 14·49-s − 30·59-s + 32·64-s − 6·71-s + 36·80-s + 16·81-s + 18·89-s + 16·100-s + 40·124-s − 3·125-s + 127-s + 131-s + 137-s + 139-s − 60·144-s + 149-s + 151-s + 30·155-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.34·5-s − 5/3·9-s + 3·16-s + 2.68·20-s + 4/5·25-s + 1.79·31-s − 3.33·36-s − 2.23·45-s + 2·49-s − 3.90·59-s + 4·64-s − 0.712·71-s + 4.02·80-s + 16/9·81-s + 1.90·89-s + 8/5·100-s + 3.59·124-s − 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5·144-s + 0.0819·149-s + 0.0813·151-s + 2.40·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.750617892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.750617892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−10.67802042359854566449215005045, −10.50365325630705075796403638387, −10.33985578465369453467663329351, −9.502301252710149187502735559205, −9.284044306472577975284911193307, −8.635997787248410027129593456426, −8.215912503114003664649786779515, −7.67500229435343576420001945342, −7.35585356436051050074478985162, −6.60521237518476153560234420747, −6.16869289395445311960212154345, −6.16849001540665366062033611885, −5.57963650838486912801322515912, −5.17754805365538913671098916388, −4.34742651288809560520636681841, −3.23506279303977129601456255998, −3.00724456626015192435405198021, −2.42410438241969137702583412487, −1.98750238801095507182588716509, −1.12835459623987926917148295459,
1.12835459623987926917148295459, 1.98750238801095507182588716509, 2.42410438241969137702583412487, 3.00724456626015192435405198021, 3.23506279303977129601456255998, 4.34742651288809560520636681841, 5.17754805365538913671098916388, 5.57963650838486912801322515912, 6.16849001540665366062033611885, 6.16869289395445311960212154345, 6.60521237518476153560234420747, 7.35585356436051050074478985162, 7.67500229435343576420001945342, 8.215912503114003664649786779515, 8.635997787248410027129593456426, 9.284044306472577975284911193307, 9.502301252710149187502735559205, 10.33985578465369453467663329351, 10.50365325630705075796403638387, 10.67802042359854566449215005045
