| L(s) = 1 | + 3·3-s + 4-s − 5·5-s + 6·9-s + 3·11-s + 3·12-s − 5·13-s − 15·15-s + 16-s − 5·17-s − 5·20-s − 23-s + 9·25-s + 9·27-s + 9·33-s + 6·36-s − 15·39-s + 3·44-s − 30·45-s + 3·48-s − 2·49-s − 15·51-s − 5·52-s + 9·53-s − 15·55-s − 15·60-s + 64-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1/2·4-s − 2.23·5-s + 2·9-s + 0.904·11-s + 0.866·12-s − 1.38·13-s − 3.87·15-s + 1/4·16-s − 1.21·17-s − 1.11·20-s − 0.208·23-s + 9/5·25-s + 1.73·27-s + 1.56·33-s + 36-s − 2.40·39-s + 0.452·44-s − 4.47·45-s + 0.433·48-s − 2/7·49-s − 2.10·51-s − 0.693·52-s + 1.23·53-s − 2.02·55-s − 1.93·60-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 438012 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 438012 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 23 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 51 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 108 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.281332014480734507686463382372, −7.79693084013738314313676529376, −7.66789484637525947828872256743, −7.18397458027831275084905593909, −6.81699084358108692939333787254, −6.39745871931626390237679545427, −5.38576310114245014632109805928, −4.69254767982085797856428267091, −4.12576730667187250286037975668, −3.98366473405145155523022042406, −3.48244561279109951222497999456, −2.71439601719013453935072567899, −2.39422209402380212499518383584, −1.45834919175506091700881127846, 0,
1.45834919175506091700881127846, 2.39422209402380212499518383584, 2.71439601719013453935072567899, 3.48244561279109951222497999456, 3.98366473405145155523022042406, 4.12576730667187250286037975668, 4.69254767982085797856428267091, 5.38576310114245014632109805928, 6.39745871931626390237679545427, 6.81699084358108692939333787254, 7.18397458027831275084905593909, 7.66789484637525947828872256743, 7.79693084013738314313676529376, 8.281332014480734507686463382372
