| L(s) = 1 | + 2-s − 4·3-s + 4-s − 4·6-s − 2·7-s + 8-s + 7·9-s − 4·11-s − 4·12-s − 2·14-s + 16-s − 6·17-s + 7·18-s + 4·19-s + 8·21-s − 4·22-s − 8·23-s − 4·24-s − 9·25-s − 4·27-s − 2·28-s + 4·29-s + 32-s + 16·33-s − 6·34-s + 7·36-s + 8·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.30·3-s + 1/2·4-s − 1.63·6-s − 0.755·7-s + 0.353·8-s + 7/3·9-s − 1.20·11-s − 1.15·12-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 1.64·18-s + 0.917·19-s + 1.74·21-s − 0.852·22-s − 1.66·23-s − 0.816·24-s − 9/5·25-s − 0.769·27-s − 0.377·28-s + 0.742·29-s + 0.176·32-s + 2.78·33-s − 1.02·34-s + 7/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−16.6572978882, −16.2105519665, −15.8682373749, −15.4425168943, −15.1526266873, −13.9215157205, −13.5922863880, −13.3506865437, −12.5054660076, −12.1499536241, −11.6433933606, −11.4971267324, −10.6973055268, −10.2802576297, −9.97691261465, −9.03952801672, −8.05229726544, −7.45985560668, −6.65408062467, −6.08119338192, −5.88908450916, −5.12564549383, −4.61153453491, −3.64849442843, −2.34351544306, 0,
2.34351544306, 3.64849442843, 4.61153453491, 5.12564549383, 5.88908450916, 6.08119338192, 6.65408062467, 7.45985560668, 8.05229726544, 9.03952801672, 9.97691261465, 10.2802576297, 10.6973055268, 11.4971267324, 11.6433933606, 12.1499536241, 12.5054660076, 13.3506865437, 13.5922863880, 13.9215157205, 15.1526266873, 15.4425168943, 15.8682373749, 16.2105519665, 16.6572978882
