| L(s) = 1 | − 4·4-s + 7-s + 3·9-s − 11-s + 8·13-s + 12·16-s − 4·17-s + 12·19-s − 10·23-s − 9·25-s − 4·28-s − 12·36-s − 10·37-s + 4·41-s + 4·44-s + 49-s − 32·52-s − 12·53-s + 4·61-s + 3·63-s − 32·64-s − 6·67-s + 16·68-s + 2·71-s − 20·73-s − 48·76-s − 77-s + ⋯ |
| L(s) = 1 | − 2·4-s + 0.377·7-s + 9-s − 0.301·11-s + 2.21·13-s + 3·16-s − 0.970·17-s + 2.75·19-s − 2.08·23-s − 9/5·25-s − 0.755·28-s − 2·36-s − 1.64·37-s + 0.624·41-s + 0.603·44-s + 1/7·49-s − 4.43·52-s − 1.64·53-s + 0.512·61-s + 0.377·63-s − 4·64-s − 0.733·67-s + 1.94·68-s + 0.237·71-s − 2.34·73-s − 5.50·76-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456533 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456533 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−8.333198695137771254103563674887, −7.956327480837723148155835866573, −7.66866498238269472650919646245, −7.17057178520211720164419084101, −6.27653205655726142636171995192, −5.80118601537640384820432803021, −5.58596676881303008604275522263, −4.95399628290113050953278162390, −4.39514744576497898862871693801, −3.85266175099960933742282200478, −3.79024025385094143595808125256, −3.04533793513601036750822233008, −1.58751343454368774869669820755, −1.32889476078570135837429482480, 0,
1.32889476078570135837429482480, 1.58751343454368774869669820755, 3.04533793513601036750822233008, 3.79024025385094143595808125256, 3.85266175099960933742282200478, 4.39514744576497898862871693801, 4.95399628290113050953278162390, 5.58596676881303008604275522263, 5.80118601537640384820432803021, 6.27653205655726142636171995192, 7.17057178520211720164419084101, 7.66866498238269472650919646245, 7.956327480837723148155835866573, 8.333198695137771254103563674887
