| L(s) = 1 | − 3·4-s + 7-s − 2·9-s + 11-s − 8·13-s + 5·16-s − 8·17-s − 8·23-s − 6·25-s − 3·28-s + 6·36-s − 12·37-s − 8·41-s − 3·44-s + 49-s + 24·52-s − 12·53-s − 2·63-s − 3·64-s + 16·67-s + 24·68-s − 24·71-s + 16·73-s + 77-s − 5·81-s − 8·91-s + 24·92-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.377·7-s − 2/3·9-s + 0.301·11-s − 2.21·13-s + 5/4·16-s − 1.94·17-s − 1.66·23-s − 6/5·25-s − 0.566·28-s + 36-s − 1.97·37-s − 1.24·41-s − 0.452·44-s + 1/7·49-s + 3.32·52-s − 1.64·53-s − 0.251·63-s − 3/8·64-s + 1.95·67-s + 2.91·68-s − 2.84·71-s + 1.87·73-s + 0.113·77-s − 5/9·81-s − 0.838·91-s + 2.50·92-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 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.339912824949481407935374648788, −7.67378178112872223010860918103, −7.37145602881524282201485228293, −6.70583207072298392541889211635, −6.25417326558836201502812232893, −5.62343675689709412781095073358, −4.94559945005359926667122742610, −4.94543315829774789302801165548, −4.28286203006294711588571475853, −3.86661484554997482031837898039, −3.17103530919462504679231815837, −2.20053354537635753960919690772, −1.91939450772296709675161614006, 0, 0,
1.91939450772296709675161614006, 2.20053354537635753960919690772, 3.17103530919462504679231815837, 3.86661484554997482031837898039, 4.28286203006294711588571475853, 4.94543315829774789302801165548, 4.94559945005359926667122742610, 5.62343675689709412781095073358, 6.25417326558836201502812232893, 6.70583207072298392541889211635, 7.37145602881524282201485228293, 7.67378178112872223010860918103, 8.339912824949481407935374648788
