| L(s) = 1 | − 2·2-s + 3·4-s − 4·5-s − 4·8-s + 6·9-s + 8·10-s + 2·11-s − 12·13-s + 5·16-s − 12·18-s − 6·19-s − 12·20-s − 4·22-s − 8·23-s + 2·25-s + 24·26-s + 12·29-s − 6·32-s + 18·36-s + 12·38-s + 16·40-s + 6·44-s − 24·45-s + 16·46-s − 16·47-s + 4·49-s − 4·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.78·5-s − 1.41·8-s + 2·9-s + 2.52·10-s + 0.603·11-s − 3.32·13-s + 5/4·16-s − 2.82·18-s − 1.37·19-s − 2.68·20-s − 0.852·22-s − 1.66·23-s + 2/5·25-s + 4.70·26-s + 2.22·29-s − 1.06·32-s + 3·36-s + 1.94·38-s + 2.52·40-s + 0.904·44-s − 3.57·45-s + 2.35·46-s − 2.33·47-s + 4/7·49-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2976551487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2976551487\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 112 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
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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
−11.61622458198183713081864071685, −10.73978347697589090123505274528, −10.37544247158519030771501800049, −9.869657420749556430021169861283, −9.812942474623324741811136710067, −9.401015651062948175712097107372, −8.404698546837973896382080257851, −8.254241647099399101113112800091, −7.71388039206146437273757543984, −7.44762265446535733044525620099, −6.95010365868690331727234701514, −6.74088728555677034406075891110, −6.02811862778622137859657311885, −4.70625924775662572057444076338, −4.64997904029601505456448773943, −4.10836519063235601385128301951, −3.34818067703718862826595783894, −2.32771420129432717051475941594, −1.84693101155797292146430206660, −0.43010836589856531599598196145,
0.43010836589856531599598196145, 1.84693101155797292146430206660, 2.32771420129432717051475941594, 3.34818067703718862826595783894, 4.10836519063235601385128301951, 4.64997904029601505456448773943, 4.70625924775662572057444076338, 6.02811862778622137859657311885, 6.74088728555677034406075891110, 6.95010365868690331727234701514, 7.44762265446535733044525620099, 7.71388039206146437273757543984, 8.254241647099399101113112800091, 8.404698546837973896382080257851, 9.401015651062948175712097107372, 9.812942474623324741811136710067, 9.869657420749556430021169861283, 10.37544247158519030771501800049, 10.73978347697589090123505274528, 11.61622458198183713081864071685
