| L(s) = 1 | − 3.92e4·3-s + 2.62e5·4-s + 1.95e6·5-s − 4.03e7·7-s + 1.15e9·9-s + 2.63e9·11-s − 1.02e10·12-s + 1.88e10·13-s − 7.67e10·15-s + 6.87e10·16-s − 5.41e10·17-s + 5.12e11·20-s + 1.58e12·21-s + 3.81e12·25-s − 3.01e13·27-s − 1.05e13·28-s − 1.46e13·29-s − 1.03e14·33-s − 7.88e13·35-s + 3.03e14·36-s − 7.39e14·39-s + 6.91e14·44-s + 2.25e15·45-s + 2.61e14·47-s − 2.69e15·48-s + 1.62e15·49-s + 2.12e15·51-s + ⋯ |
| L(s) = 1 | − 1.99·3-s + 4-s + 5-s − 7-s + 2.98·9-s + 1.11·11-s − 1.99·12-s + 1.77·13-s − 1.99·15-s + 16-s − 0.456·17-s + 20-s + 1.99·21-s + 25-s − 3.95·27-s − 28-s − 1.00·29-s − 2.23·33-s − 35-s + 2.98·36-s − 3.54·39-s + 1.11·44-s + 2.98·45-s + 0.233·47-s − 1.99·48-s + 49-s + 0.911·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(1.908910046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.908910046\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{9} T \) |
| 7 | \( 1 + p^{9} T \) |
| good | 2 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 3 | \( 1 + 39286 T + p^{18} T^{2} \) |
| 11 | \( 1 - 2637510122 T + p^{18} T^{2} \) |
| 13 | \( 1 - 18822563674 T + p^{18} T^{2} \) |
| 17 | \( 1 + 54170520014 T + p^{18} T^{2} \) |
| 19 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 23 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 29 | \( 1 + 14623240000822 T + p^{18} T^{2} \) |
| 31 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 37 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 41 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 43 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 47 | \( 1 - 261032556546466 T + p^{18} T^{2} \) |
| 53 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 59 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 61 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 67 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 71 | \( 1 - 11592841444168322 T + p^{18} T^{2} \) |
| 73 | \( 1 - 100714945072821154 T + p^{18} T^{2} \) |
| 79 | \( 1 + 126856003456085902 T + p^{18} T^{2} \) |
| 83 | \( 1 - 367224638997840874 T + p^{18} T^{2} \) |
| 89 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 97 | \( 1 + 1519865703804919214 T + p^{18} T^{2} \) |
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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
−12.42459793701292601270519829565, −11.30779956300603346008085835612, −10.61236753277556951375079821213, −9.462956076554331081450849108357, −6.83515174511980189418696504660, −6.31343685893712754514652474143, −5.64330879812955233604340869914, −3.79955597604356537769931674123, −1.71417844707844582023418249395, −0.848952803579234635190800726275,
0.848952803579234635190800726275, 1.71417844707844582023418249395, 3.79955597604356537769931674123, 5.64330879812955233604340869914, 6.31343685893712754514652474143, 6.83515174511980189418696504660, 9.462956076554331081450849108357, 10.61236753277556951375079821213, 11.30779956300603346008085835612, 12.42459793701292601270519829565
