| L(s) = 1 | + 2·3-s − 2·4-s − 2·5-s + 2·7-s − 3·9-s + 2·11-s − 4·12-s + 6·13-s − 4·15-s + 4·16-s + 6·17-s + 8·19-s + 4·20-s + 4·21-s + 3·25-s − 14·27-s − 4·28-s − 6·29-s − 4·31-s + 4·33-s − 4·35-s + 6·36-s + 8·37-s + 12·39-s − 4·41-s − 16·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 0.894·5-s + 0.755·7-s − 9-s + 0.603·11-s − 1.15·12-s + 1.66·13-s − 1.03·15-s + 16-s + 1.45·17-s + 1.83·19-s + 0.894·20-s + 0.872·21-s + 3/5·25-s − 2.69·27-s − 0.755·28-s − 1.11·29-s − 0.718·31-s + 0.696·33-s − 0.676·35-s + 36-s + 1.31·37-s + 1.92·39-s − 0.624·41-s − 2.43·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.259913462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.259913462\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−15.3832112075, −15.2725263569, −14.6425906738, −14.2620991577, −14.0962101507, −13.3702303857, −13.3483260723, −12.3253573140, −11.8074747577, −11.4484940335, −11.0981108270, −10.2617561341, −9.46097475146, −9.18445227238, −8.67949885325, −8.05573815478, −7.98094729632, −7.40499721789, −6.21937523368, −5.47907564041, −5.14802755063, −3.84388371347, −3.61689003046, −3.05770384157, −1.34060858351,
1.34060858351, 3.05770384157, 3.61689003046, 3.84388371347, 5.14802755063, 5.47907564041, 6.21937523368, 7.40499721789, 7.98094729632, 8.05573815478, 8.67949885325, 9.18445227238, 9.46097475146, 10.2617561341, 11.0981108270, 11.4484940335, 11.8074747577, 12.3253573140, 13.3483260723, 13.3702303857, 14.0962101507, 14.2620991577, 14.6425906738, 15.2725263569, 15.3832112075
