| L(s) = 1 | + 4·5-s + 9-s − 8·11-s + 4·13-s + 2·25-s + 16·31-s − 8·43-s + 4·45-s − 7·49-s − 32·55-s + 4·61-s + 16·65-s + 8·67-s + 81-s − 8·99-s + 36·101-s + 32·103-s + 24·107-s + 36·113-s + 4·117-s + 26·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s − 32·143-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1/3·9-s − 2.41·11-s + 1.10·13-s + 2/5·25-s + 2.87·31-s − 1.21·43-s + 0.596·45-s − 49-s − 4.31·55-s + 0.512·61-s + 1.98·65-s + 0.977·67-s + 1/9·81-s − 0.804·99-s + 3.58·101-s + 3.15·103-s + 2.32·107-s + 3.38·113-s + 0.369·117-s + 2.36·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.67·143-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.485828742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.485828742\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.456614023795656741730916500766, −8.304042274266532161103040174092, −7.61947548271568794045481490015, −7.35977874808944660694472335106, −6.41480629060511724242965007307, −6.16826675668887144123811676692, −5.98894980010644060477023411018, −5.14444399128618341481533912587, −5.02364207902997377499660058417, −4.44992200701182918033448908949, −3.43299575130251299450107440988, −3.00889317555517420664706578512, −2.17837496002353596178446785744, −1.99999348241168201324552330088, −0.852162344962028721020593862686,
0.852162344962028721020593862686, 1.99999348241168201324552330088, 2.17837496002353596178446785744, 3.00889317555517420664706578512, 3.43299575130251299450107440988, 4.44992200701182918033448908949, 5.02364207902997377499660058417, 5.14444399128618341481533912587, 5.98894980010644060477023411018, 6.16826675668887144123811676692, 6.41480629060511724242965007307, 7.35977874808944660694472335106, 7.61947548271568794045481490015, 8.304042274266532161103040174092, 8.456614023795656741730916500766
