| L(s) = 1 | + 2·3-s − 4-s + 6·7-s + 3·9-s − 2·12-s + 16-s − 2·17-s − 10·19-s + 12·21-s + 8·23-s + 4·27-s − 6·28-s − 3·36-s − 14·37-s + 2·48-s + 13·49-s − 4·51-s − 20·57-s + 18·63-s − 64-s + 2·68-s + 16·69-s − 32·73-s + 10·76-s + 5·81-s − 12·84-s + 20·89-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 2.26·7-s + 9-s − 0.577·12-s + 1/4·16-s − 0.485·17-s − 2.29·19-s + 2.61·21-s + 1.66·23-s + 0.769·27-s − 1.13·28-s − 1/2·36-s − 2.30·37-s + 0.288·48-s + 13/7·49-s − 0.560·51-s − 2.64·57-s + 2.26·63-s − 1/8·64-s + 0.242·68-s + 1.92·69-s − 3.74·73-s + 1.14·76-s + 5/9·81-s − 1.30·84-s + 2.11·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.331373102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.331373102\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.814492611253236762419670178101, −8.758628730543392213761151712964, −8.553616269416643031262354882276, −8.045918537814027003161005333570, −7.60912447234904777929343452651, −7.38827004443361711874491329717, −7.00068534104750459815646256204, −6.35173617493262002993776868021, −6.15932576651683631512431822203, −5.19417310372144516940697461133, −5.14638721457110529642630607251, −4.67003097759399706220758884006, −4.33218240136048952965833657583, −4.05386416612828761843093589297, −3.32069051575436345677288124251, −2.99058271011524248995404386867, −2.21729136626965910110451917629, −1.72805599395958158679854168062, −1.69391109285674362776025887889, −0.63230876336020523563988204071,
0.63230876336020523563988204071, 1.69391109285674362776025887889, 1.72805599395958158679854168062, 2.21729136626965910110451917629, 2.99058271011524248995404386867, 3.32069051575436345677288124251, 4.05386416612828761843093589297, 4.33218240136048952965833657583, 4.67003097759399706220758884006, 5.14638721457110529642630607251, 5.19417310372144516940697461133, 6.15932576651683631512431822203, 6.35173617493262002993776868021, 7.00068534104750459815646256204, 7.38827004443361711874491329717, 7.60912447234904777929343452651, 8.045918537814027003161005333570, 8.553616269416643031262354882276, 8.758628730543392213761151712964, 8.814492611253236762419670178101
