L(s) = 1 | + 8·5-s − 7-s + 9-s + 4·11-s − 12·13-s + 38·25-s − 8·35-s − 8·43-s + 8·45-s + 24·47-s + 49-s + 32·55-s + 12·61-s − 63-s − 96·65-s − 16·67-s − 4·77-s + 81-s + 12·91-s + 4·99-s + 32·101-s − 32·103-s + 36·107-s + 20·113-s − 12·117-s − 10·121-s + 136·125-s + ⋯ |
L(s) = 1 | + 3.57·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 3.32·13-s + 38/5·25-s − 1.35·35-s − 1.21·43-s + 1.19·45-s + 3.50·47-s + 1/7·49-s + 4.31·55-s + 1.53·61-s − 0.125·63-s − 11.9·65-s − 1.95·67-s − 0.455·77-s + 1/9·81-s + 1.25·91-s + 0.402·99-s + 3.18·101-s − 3.15·103-s + 3.48·107-s + 1.88·113-s − 1.10·117-s − 0.909·121-s + 12.1·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.043907586\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.043907586\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−8.557246747639740629686095607145, −7.48055140088893310165370888362, −7.16129745430315884399569933958, −6.96207355783957131001717701468, −6.29720922441833452882432131874, −6.05700480174216171276936327443, −5.57231195325264994616573262274, −5.14056436828631480262325484346, −4.80916909916633799382690392302, −4.24974945413085103790315507050, −3.18166849287238476303908403717, −2.53627159495471071974969043910, −2.24239796255929817450421188994, −1.86729050078829565643852869820, −0.995983380836290695648376877184,
0.995983380836290695648376877184, 1.86729050078829565643852869820, 2.24239796255929817450421188994, 2.53627159495471071974969043910, 3.18166849287238476303908403717, 4.24974945413085103790315507050, 4.80916909916633799382690392302, 5.14056436828631480262325484346, 5.57231195325264994616573262274, 6.05700480174216171276936327443, 6.29720922441833452882432131874, 6.96207355783957131001717701468, 7.16129745430315884399569933958, 7.48055140088893310165370888362, 8.557246747639740629686095607145