| L(s) = 1 | − 4-s − 4·7-s − 6·9-s − 8·11-s + 16-s − 25-s + 4·28-s + 6·36-s − 12·37-s − 4·41-s + 8·44-s + 12·47-s − 2·49-s + 24·53-s + 24·63-s − 64-s + 32·67-s − 24·71-s + 12·73-s + 32·77-s + 27·81-s − 24·83-s + 48·99-s + 100-s − 12·101-s − 4·112-s + 26·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.51·7-s − 2·9-s − 2.41·11-s + 1/4·16-s − 1/5·25-s + 0.755·28-s + 36-s − 1.97·37-s − 0.624·41-s + 1.20·44-s + 1.75·47-s − 2/7·49-s + 3.29·53-s + 3.02·63-s − 1/8·64-s + 3.90·67-s − 2.84·71-s + 1.40·73-s + 3.64·77-s + 3·81-s − 2.63·83-s + 4.82·99-s + 1/10·100-s − 1.19·101-s − 0.377·112-s + 2.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2334107919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2334107919\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 37 | $C_2$ | \( 1 + 12 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−12.03826810028248627038594676610, −11.07109260805267999150071529809, −10.60869895369719926126145627391, −10.24167447216974701997851598360, −9.924398957555058572571276514208, −9.303231176338774140092169937991, −8.783755311382713356914365949788, −8.276123716940009632573017156543, −8.270211059726016391627644799615, −7.29452499966258763890898281695, −7.01182906504780353175570139737, −6.20481603521064732086185485013, −5.69240845477110228467124213570, −5.32443839121038741741428056907, −5.06663015609205942721985426071, −3.85994058564767420354237173532, −3.42632505310477358348599745520, −2.60079527194024204754396335763, −2.53065854827239896551689471753, −0.30476611496144062866230422914,
0.30476611496144062866230422914, 2.53065854827239896551689471753, 2.60079527194024204754396335763, 3.42632505310477358348599745520, 3.85994058564767420354237173532, 5.06663015609205942721985426071, 5.32443839121038741741428056907, 5.69240845477110228467124213570, 6.20481603521064732086185485013, 7.01182906504780353175570139737, 7.29452499966258763890898281695, 8.270211059726016391627644799615, 8.276123716940009632573017156543, 8.783755311382713356914365949788, 9.303231176338774140092169937991, 9.924398957555058572571276514208, 10.24167447216974701997851598360, 10.60869895369719926126145627391, 11.07109260805267999150071529809, 12.03826810028248627038594676610
