| L(s) = 1 | − 2·2-s − 4·3-s + 4-s + 8·6-s − 8·7-s + 2·8-s + 6·9-s − 4·11-s − 4·12-s + 16·14-s − 3·16-s − 4·17-s − 12·18-s − 4·19-s + 32·21-s + 8·22-s − 8·24-s + 25-s + 4·27-s − 8·28-s − 4·29-s − 8·31-s − 2·32-s + 16·33-s + 8·34-s + 6·36-s + 8·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s + 3.26·6-s − 3.02·7-s + 0.707·8-s + 2·9-s − 1.20·11-s − 1.15·12-s + 4.27·14-s − 3/4·16-s − 0.970·17-s − 2.82·18-s − 0.917·19-s + 6.98·21-s + 1.70·22-s − 1.63·24-s + 1/5·25-s + 0.769·27-s − 1.51·28-s − 0.742·29-s − 1.43·31-s − 0.353·32-s + 2.78·33-s + 1.37·34-s + 36-s + 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 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
−17.3405617440, −16.6814250399, −16.6216229554, −16.0534712890, −15.9784997767, −15.2532327675, −14.3654180706, −13.2244479880, −12.9625945798, −12.9146361946, −12.1373187793, −11.4946665500, −10.7527712458, −10.6512482270, −9.88449909181, −9.84497658306, −8.74716016951, −8.67915048394, −7.13284046688, −7.04231773997, −6.21865412034, −5.80677556176, −5.20912025980, −4.03550878859, −2.83582044678, 0, 0,
2.83582044678, 4.03550878859, 5.20912025980, 5.80677556176, 6.21865412034, 7.04231773997, 7.13284046688, 8.67915048394, 8.74716016951, 9.84497658306, 9.88449909181, 10.6512482270, 10.7527712458, 11.4946665500, 12.1373187793, 12.9146361946, 12.9625945798, 13.2244479880, 14.3654180706, 15.2532327675, 15.9784997767, 16.0534712890, 16.6216229554, 16.6814250399, 17.3405617440
