| L(s) = 1 | − 4·3-s − 3·4-s − 6·5-s + 6·9-s − 5·11-s + 12·12-s + 24·15-s + 5·16-s + 18·20-s − 18·23-s + 17·25-s + 4·27-s + 20·33-s − 18·36-s + 16·37-s + 15·44-s − 36·45-s + 8·47-s − 20·48-s − 13·49-s + 12·53-s + 30·55-s + 18·59-s − 72·60-s − 3·64-s − 14·67-s + 72·69-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3/2·4-s − 2.68·5-s + 2·9-s − 1.50·11-s + 3.46·12-s + 6.19·15-s + 5/4·16-s + 4.02·20-s − 3.75·23-s + 17/5·25-s + 0.769·27-s + 3.48·33-s − 3·36-s + 2.63·37-s + 2.26·44-s − 5.36·45-s + 1.16·47-s − 2.88·48-s − 1.85·49-s + 1.64·53-s + 4.04·55-s + 2.34·59-s − 9.29·60-s − 3/8·64-s − 1.71·67-s + 8.66·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450241 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.214644259662643725428835431279, −7.66039763333771123227596439320, −7.44527095540533686848888098295, −6.64444629781009788738361946627, −6.00646746388851025333489625692, −5.60157612088078313975836309422, −5.43650672338497627064179356489, −4.51604888222126848033561730521, −4.26508467002293325004309474944, −4.25666915587159511841100490901, −3.43011094815890270849961379878, −2.55202902956317307533327873816, −0.75342584052823796311141171419, 0, 0,
0.75342584052823796311141171419, 2.55202902956317307533327873816, 3.43011094815890270849961379878, 4.25666915587159511841100490901, 4.26508467002293325004309474944, 4.51604888222126848033561730521, 5.43650672338497627064179356489, 5.60157612088078313975836309422, 6.00646746388851025333489625692, 6.64444629781009788738361946627, 7.44527095540533686848888098295, 7.66039763333771123227596439320, 8.214644259662643725428835431279
