| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s − 4·7-s − 8-s − 2·9-s − 10-s + 2·11-s + 2·13-s + 4·14-s − 15-s − 16-s − 5·17-s + 2·18-s − 3·19-s + 4·21-s − 2·22-s − 3·23-s + 24-s − 25-s − 2·26-s + 2·27-s − 2·29-s + 30-s − 2·31-s + 6·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.603·11-s + 0.554·13-s + 1.06·14-s − 0.258·15-s − 1/4·16-s − 1.21·17-s + 0.471·18-s − 0.688·19-s + 0.872·21-s − 0.426·22-s − 0.625·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.384·27-s − 0.371·29-s + 0.182·30-s − 0.359·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10137 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10137 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 109 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 9 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T - 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 51 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 17 T + 180 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 116 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 74 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 53 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−17.1217153272, −16.2709489158, −16.0803424531, −15.5052709784, −15.0023014457, −14.3350405922, −13.6664271261, −13.3814302256, −12.8139563545, −12.2575516314, −11.6763781273, −11.1642780379, −10.6818979264, −9.97041136450, −9.46928791595, −9.08154722630, −8.62625935954, −7.92976704737, −6.75091706142, −6.47751928467, −6.09764155755, −5.28050615602, −4.18122217263, −3.35403929419, −2.24163739117, 0,
2.24163739117, 3.35403929419, 4.18122217263, 5.28050615602, 6.09764155755, 6.47751928467, 6.75091706142, 7.92976704737, 8.62625935954, 9.08154722630, 9.46928791595, 9.97041136450, 10.6818979264, 11.1642780379, 11.6763781273, 12.2575516314, 12.8139563545, 13.3814302256, 13.6664271261, 14.3350405922, 15.0023014457, 15.5052709784, 16.0803424531, 16.2709489158, 17.1217153272
