L(s) = 1 | + 3.93e4·3-s − 5.24e5·4-s + 1.16e9·9-s − 2.06e10·12-s + 7.19e9·13-s + 2.06e11·16-s − 7.62e12·25-s + 3.05e13·27-s − 6.09e14·36-s + 2.83e14·39-s − 1.46e15·43-s + 8.11e15·48-s + 2.75e15·49-s − 3.77e15·52-s + 1.89e16·61-s − 7.20e16·64-s − 3.00e17·75-s − 2.81e17·79-s + 7.50e17·81-s + 4.00e18·100-s − 5.21e18·103-s − 1.59e19·108-s + 8.36e18·117-s − 1.11e19·121-s + 127-s − 5.75e19·129-s + 131-s + ⋯ |
L(s) = 1 | + 2·3-s − 2·4-s + 3·9-s − 4·12-s + 0.678·13-s + 3·16-s − 2·25-s + 4·27-s − 6·36-s + 1.35·39-s − 2.90·43-s + 6·48-s + 1.69·49-s − 1.35·52-s + 1.62·61-s − 4·64-s − 4·75-s − 2.34·79-s + 5·81-s + 4·100-s − 3.99·103-s − 8·108-s + 2.03·117-s − 2·121-s − 5.81·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+9)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{19}{2})\) |
\(\approx\) |
\(2.952865025\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.952865025\) |
\(L(10)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( ( 1 - p^{9} T )^{2} \) |
| 13 | $C_2$ | \( 1 - 7197541846 T + p^{18} T^{2} \) |
good | 2 | $C_2$ | \( ( 1 + p^{18} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p^{18} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 77549186 T + p^{18} T^{2} )( 1 + 77549186 T + p^{18} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{18} T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{9} T )^{2}( 1 + p^{9} T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 308559680858 T + p^{18} T^{2} )( 1 + 308559680858 T + p^{18} T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{9} T )^{2}( 1 + p^{9} T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{9} T )^{2}( 1 + p^{9} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 50018992173358 T + p^{18} T^{2} )( 1 + 50018992173358 T + p^{18} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 23240947030054 T + p^{18} T^{2} )( 1 + 23240947030054 T + p^{18} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{18} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 730385642547286 T + p^{18} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{18} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{9} T )^{2}( 1 + p^{9} T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{18} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 9487161099916918 T + p^{18} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 41747295001607494 T + p^{18} T^{2} )( 1 + 41747295001607494 T + p^{18} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{18} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 29908998244279726 T + p^{18} T^{2} )( 1 + 29908998244279726 T + p^{18} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 140655567501204338 T + p^{18} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{18} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{18} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 140873967896062466 T + p^{18} T^{2} )( 1 + 140873967896062466 T + p^{18} T^{2} ) \) |
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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
−13.15298627470623359341446478666, −12.44333287379212321129771001903, −11.72438028062121135979584031833, −10.43675031828537215297964601643, −9.960000200394304511503400107175, −9.607674717835733369238417731210, −9.029895052131635350987039983304, −8.354056994453547721184981004849, −8.321663605523985923545252663966, −7.60801394350844151101269361171, −6.80216166738754828287703267536, −5.70310285243492338971731032160, −5.02064758495337757302927168666, −4.14505679276865527834955683370, −3.94809117774332293775672768667, −3.41270226914847917844602319010, −2.66863480659590350305211206742, −1.69210466534197031194707738663, −1.26685120018441289932379391096, −0.37405293840099820245824893154,
0.37405293840099820245824893154, 1.26685120018441289932379391096, 1.69210466534197031194707738663, 2.66863480659590350305211206742, 3.41270226914847917844602319010, 3.94809117774332293775672768667, 4.14505679276865527834955683370, 5.02064758495337757302927168666, 5.70310285243492338971731032160, 6.80216166738754828287703267536, 7.60801394350844151101269361171, 8.321663605523985923545252663966, 8.354056994453547721184981004849, 9.029895052131635350987039983304, 9.607674717835733369238417731210, 9.960000200394304511503400107175, 10.43675031828537215297964601643, 11.72438028062121135979584031833, 12.44333287379212321129771001903, 13.15298627470623359341446478666