| L(s) = 1 | + 5.90e4·3-s − 7.73e7·7-s + 3.48e9·9-s + 2.17e11·13-s − 3.84e12·19-s − 4.56e12·21-s + 9.53e13·25-s + 2.05e14·27-s + 7.93e14·31-s + 8.66e15·37-s + 1.28e16·39-s + 3.67e16·43-s − 7.38e16·49-s − 2.26e17·57-s − 1.38e18·61-s − 2.69e17·63-s − 1.57e18·67-s + 8.57e18·73-s + 5.63e18·75-s − 3.26e18·79-s + 1.21e19·81-s − 1.68e19·91-s + 4.68e19·93-s − 1.46e20·97-s − 2.57e20·103-s − 1.61e20·109-s + 5.11e20·111-s + ⋯ |
| L(s) = 1 | + 3-s − 0.273·7-s + 9-s + 1.57·13-s − 0.626·19-s − 0.273·21-s + 25-s + 27-s + 0.968·31-s + 1.80·37-s + 1.57·39-s + 1.70·43-s − 0.925·49-s − 0.626·57-s − 1.94·61-s − 0.273·63-s − 0.866·67-s + 1.99·73-s + 75-s − 0.344·79-s + 81-s − 0.431·91-s + 0.968·93-s − 1.98·97-s − 1.91·103-s − 0.681·109-s + 1.80·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(3.100873455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.100873455\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{10} T \) |
| good | 5 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 7 | \( 1 + 77335774 T + p^{20} T^{2} \) |
| 11 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 13 | \( 1 - 217393135826 T + p^{20} T^{2} \) |
| 17 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 19 | \( 1 + 3843000838126 T + p^{20} T^{2} \) |
| 23 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 29 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 31 | \( 1 - 793864193940674 T + p^{20} T^{2} \) |
| 37 | \( 1 - 8665815522315698 T + p^{20} T^{2} \) |
| 41 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 43 | \( 1 - 36742119154315826 T + p^{20} T^{2} \) |
| 47 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 53 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 59 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 61 | \( 1 + 1387788230779990126 T + p^{20} T^{2} \) |
| 67 | \( 1 + 1579433466595168174 T + p^{20} T^{2} \) |
| 71 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 73 | \( 1 - 8577821547816235298 T + p^{20} T^{2} \) |
| 79 | \( 1 + 3262829215661625598 T + p^{20} T^{2} \) |
| 83 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 89 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 97 | \( 1 + \)\(14\!\cdots\!74\)\( T + p^{20} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.17912925631463923982994415669, −13.84188662942064646240882346636, −12.76627429144628553066711298825, −10.80291062586455519651544853193, −9.241209741726552886100043064313, −8.074705568652397708123583786715, −6.38947847790801941231934800710, −4.18921653788178283192069206214, −2.81822739497257555948787993011, −1.16533056067096447947041386933,
1.16533056067096447947041386933, 2.81822739497257555948787993011, 4.18921653788178283192069206214, 6.38947847790801941231934800710, 8.074705568652397708123583786715, 9.241209741726552886100043064313, 10.80291062586455519651544853193, 12.76627429144628553066711298825, 13.84188662942064646240882346636, 15.17912925631463923982994415669
