| L(s) = 1 | + 2·2-s − 4-s − 7-s − 8·8-s − 2·9-s + 2·11-s − 2·14-s − 7·16-s − 4·18-s + 4·22-s − 8·23-s − 6·25-s + 28-s − 12·29-s + 14·32-s + 2·36-s − 12·37-s + 24·43-s − 2·44-s − 16·46-s + 49-s − 12·50-s − 12·53-s + 8·56-s − 24·58-s + 2·63-s + 35·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 0.377·7-s − 2.82·8-s − 2/3·9-s + 0.603·11-s − 0.534·14-s − 7/4·16-s − 0.942·18-s + 0.852·22-s − 1.66·23-s − 6/5·25-s + 0.188·28-s − 2.22·29-s + 2.47·32-s + 1/3·36-s − 1.97·37-s + 3.65·43-s − 0.301·44-s − 2.35·46-s + 1/7·49-s − 1.69·50-s − 1.64·53-s + 1.06·56-s − 3.15·58-s + 0.251·63-s + 35/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41503 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41503 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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
−9.757325112612361159806534316063, −9.519898606480484632296989725464, −8.920762498226765508675400216409, −8.678915900547504955372461266927, −7.82867506007281568614416512666, −7.37145602881524282201485228293, −6.25417326558836201502812232893, −5.96443978110022033983848919969, −5.62343675689709412781095073358, −4.94543315829774789302801165548, −4.07930883034575373640745113861, −3.86661484554997482031837898039, −3.26888344602981049454952325554, −2.20053354537635753960919690772, 0,
2.20053354537635753960919690772, 3.26888344602981049454952325554, 3.86661484554997482031837898039, 4.07930883034575373640745113861, 4.94543315829774789302801165548, 5.62343675689709412781095073358, 5.96443978110022033983848919969, 6.25417326558836201502812232893, 7.37145602881524282201485228293, 7.82867506007281568614416512666, 8.678915900547504955372461266927, 8.920762498226765508675400216409, 9.519898606480484632296989725464, 9.757325112612361159806534316063
