| L(s) = 1 | − 2-s − 3-s − 3·5-s + 6-s − 7-s + 8-s + 3·10-s − 3·11-s + 2·13-s + 14-s + 3·15-s − 16-s − 6·17-s + 4·19-s + 21-s + 3·22-s − 6·23-s − 24-s + 5·25-s − 2·26-s + 27-s − 18·29-s − 3·30-s − 5·31-s + 3·33-s + 6·34-s + 3·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.948·10-s − 0.904·11-s + 0.554·13-s + 0.267·14-s + 0.774·15-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.218·21-s + 0.639·22-s − 1.25·23-s − 0.204·24-s + 25-s − 0.392·26-s + 0.192·27-s − 3.34·29-s − 0.547·30-s − 0.898·31-s + 0.522·33-s + 1.02·34-s + 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−10.61434200521800674124806547370, −10.29371147409672592546779862741, −9.530317972917441520281070678393, −9.413093394721318902564652020265, −8.633890351343381290377276780713, −8.467410706397419159088658096491, −7.902265035284963014808339096881, −7.49951177633371495155973191650, −7.08427286563210166489124730528, −6.67264002191691540555359713397, −5.98376780161361039196571721392, −5.28177361706545084344224758059, −5.19647863667798106517169958182, −4.26041074923174940071755008969, −3.66907189885344088449038057072, −3.53513837210955347214619389865, −2.37758973051070315587278182700, −1.62430535538736806509312317435, 0, 0,
1.62430535538736806509312317435, 2.37758973051070315587278182700, 3.53513837210955347214619389865, 3.66907189885344088449038057072, 4.26041074923174940071755008969, 5.19647863667798106517169958182, 5.28177361706545084344224758059, 5.98376780161361039196571721392, 6.67264002191691540555359713397, 7.08427286563210166489124730528, 7.49951177633371495155973191650, 7.902265035284963014808339096881, 8.467410706397419159088658096491, 8.633890351343381290377276780713, 9.413093394721318902564652020265, 9.530317972917441520281070678393, 10.29371147409672592546779862741, 10.61434200521800674124806547370
