| L(s) = 1 | − 4-s + 4·5-s + 6·9-s + 6·11-s + 16-s − 10·19-s − 4·20-s + 11·25-s + 8·29-s + 4·31-s − 6·36-s − 6·41-s − 6·44-s + 24·45-s + 24·55-s − 8·59-s − 12·61-s − 64-s − 12·71-s + 10·76-s − 28·79-s + 4·80-s + 27·81-s + 4·89-s − 40·95-s + 36·99-s − 11·100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s + 2·9-s + 1.80·11-s + 1/4·16-s − 2.29·19-s − 0.894·20-s + 11/5·25-s + 1.48·29-s + 0.718·31-s − 36-s − 0.937·41-s − 0.904·44-s + 3.57·45-s + 3.23·55-s − 1.04·59-s − 1.53·61-s − 1/8·64-s − 1.42·71-s + 1.14·76-s − 3.15·79-s + 0.447·80-s + 3·81-s + 0.423·89-s − 4.10·95-s + 3.61·99-s − 1.09·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.898939002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.898939002\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−10.80507070742835426501614440460, −10.66035997935490810962620368713, −10.11853796096329643550247952201, −9.955802034071169369266024745726, −9.396151272125967098321882505444, −9.145893820676618331685318581887, −8.544915215203032126277274070988, −8.345356990409241166031175342836, −7.33690888757180750146905209485, −6.91510649035026153117224646481, −6.46620140662974376235907547777, −6.25517366307462395317932108514, −5.74452128344921372796566132051, −4.69355457239090258952234151741, −4.55356671093030157894897454841, −4.18123539548456914070844542157, −3.30463564635477408817207360554, −2.38448690973841340693879134936, −1.55411791066255474757878657474, −1.33983583697781415895254106116,
1.33983583697781415895254106116, 1.55411791066255474757878657474, 2.38448690973841340693879134936, 3.30463564635477408817207360554, 4.18123539548456914070844542157, 4.55356671093030157894897454841, 4.69355457239090258952234151741, 5.74452128344921372796566132051, 6.25517366307462395317932108514, 6.46620140662974376235907547777, 6.91510649035026153117224646481, 7.33690888757180750146905209485, 8.345356990409241166031175342836, 8.544915215203032126277274070988, 9.145893820676618331685318581887, 9.396151272125967098321882505444, 9.955802034071169369266024745726, 10.11853796096329643550247952201, 10.66035997935490810962620368713, 10.80507070742835426501614440460
