| L(s) = 1 | + 2·5-s − 9-s − 8·11-s − 25-s + 12·29-s + 8·31-s − 20·41-s − 2·45-s − 2·49-s − 16·55-s − 8·59-s + 4·61-s + 24·79-s + 81-s + 20·89-s + 8·99-s − 4·101-s + 4·109-s + 26·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1/3·9-s − 2.41·11-s − 1/5·25-s + 2.22·29-s + 1.43·31-s − 3.12·41-s − 0.298·45-s − 2/7·49-s − 2.15·55-s − 1.04·59-s + 0.512·61-s + 2.70·79-s + 1/9·81-s + 2.11·89-s + 0.804·99-s − 0.398·101-s + 0.383·109-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7870182283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7870182283\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−15.20632694967217143748283721071, −15.13529243314735464060640336789, −13.93215941391317470745755448236, −13.69483881127454116103742377418, −13.42499414764942110476843074071, −12.64154176447702888313696528389, −12.10184840874469074539954273416, −11.46631660381950947840341031494, −10.41140256932189427651111766940, −10.39355477557200835705975196955, −9.862031298307833742750515813661, −8.890680176421272534017816198129, −8.145420643870246352308910942716, −7.88773361047321821012866564938, −6.70923995105228769594070884168, −6.16189212507377429778945636624, −5.11991543003276996574657604229, −4.92017918035329438398648872326, −3.16612952657833741339887507090, −2.33534661767996088087914166220,
2.33534661767996088087914166220, 3.16612952657833741339887507090, 4.92017918035329438398648872326, 5.11991543003276996574657604229, 6.16189212507377429778945636624, 6.70923995105228769594070884168, 7.88773361047321821012866564938, 8.145420643870246352308910942716, 8.890680176421272534017816198129, 9.862031298307833742750515813661, 10.39355477557200835705975196955, 10.41140256932189427651111766940, 11.46631660381950947840341031494, 12.10184840874469074539954273416, 12.64154176447702888313696528389, 13.42499414764942110476843074071, 13.69483881127454116103742377418, 13.93215941391317470745755448236, 15.13529243314735464060640336789, 15.20632694967217143748283721071
