| L(s) = 1 | − 2-s − 4·3-s − 4-s − 2·5-s + 4·6-s − 2·7-s + 3·8-s + 6·9-s + 2·10-s + 2·11-s + 4·12-s + 13-s + 2·14-s + 8·15-s − 16-s − 4·17-s − 6·18-s − 10·19-s + 2·20-s + 8·21-s − 2·22-s − 12·24-s + 3·25-s − 26-s + 4·27-s + 2·28-s + 8·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s − 1/2·4-s − 0.894·5-s + 1.63·6-s − 0.755·7-s + 1.06·8-s + 2·9-s + 0.632·10-s + 0.603·11-s + 1.15·12-s + 0.277·13-s + 0.534·14-s + 2.06·15-s − 1/4·16-s − 0.970·17-s − 1.41·18-s − 2.29·19-s + 0.447·20-s + 1.74·21-s − 0.426·22-s − 2.44·24-s + 3/5·25-s − 0.196·26-s + 0.769·27-s + 0.377·28-s + 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−19.6035760866, −19.2964872165, −18.4590779879, −18.1271877078, −17.3405617440, −17.3081878781, −16.6814250399, −16.0995603957, −15.9784997767, −14.7919331095, −14.3654180706, −13.0023664158, −12.9146361946, −12.1373187793, −11.5085445320, −10.8347395030, −10.7527712458, −9.84497658306, −8.67915048394, −8.55221755078, −7.04231773997, −6.57891116466, −5.80677556176, −4.78130792718, −4.03550878859, 0,
4.03550878859, 4.78130792718, 5.80677556176, 6.57891116466, 7.04231773997, 8.55221755078, 8.67915048394, 9.84497658306, 10.7527712458, 10.8347395030, 11.5085445320, 12.1373187793, 12.9146361946, 13.0023664158, 14.3654180706, 14.7919331095, 15.9784997767, 16.0995603957, 16.6814250399, 17.3081878781, 17.3405617440, 18.1271877078, 18.4590779879, 19.2964872165, 19.6035760866
