| L(s) = 1 | − 16·4-s + 62·5-s + 480·13-s + 256·16-s − 322·17-s − 992·20-s + 1.92e3·25-s + 1.76e3·29-s − 482·37-s − 4.47e3·41-s − 7.68e3·52-s + 9.59e3·61-s − 4.09e3·64-s + 2.97e4·65-s + 5.15e3·68-s − 9.11e3·73-s + 1.58e4·80-s − 6.56e3·81-s − 1.99e4·85-s + 2.49e4·89-s − 1.68e4·97-s − 3.07e4·100-s − 1.58e4·101-s − 3.12e4·109-s + 3.13e4·113-s − 2.81e4·116-s + 3.87e4·125-s + ⋯ |
| L(s) = 1 | − 4-s + 2.47·5-s + 2.84·13-s + 16-s − 1.11·17-s − 2.47·20-s + 3.07·25-s + 2.09·29-s − 0.352·37-s − 2.66·41-s − 2.84·52-s + 2.57·61-s − 64-s + 7.04·65-s + 1.11·68-s − 1.71·73-s + 2.47·80-s − 81-s − 2.76·85-s + 3.15·89-s − 1.78·97-s − 3.07·100-s − 1.55·101-s − 2.62·109-s + 2.45·113-s − 2.09·116-s + 2.47·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.083824314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083824314\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 17 | $C_2$ | \( 1 + 322 T + p^{4} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 48 T + p^{4} T^{2} )( 1 - 14 T + p^{4} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 240 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 1680 T + p^{4} T^{2} )( 1 - 82 T + p^{4} T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 1680 T + p^{4} T^{2} )( 1 + 2162 T + p^{4} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 1440 T + p^{4} T^{2} )( 1 + 3038 T + p^{4} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2482 T + p^{4} T^{2} )( 1 + 2482 T + p^{4} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6958 T + p^{4} T^{2} )( 1 - 2640 T + p^{4} T^{2} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 1442 T + p^{4} T^{2} )( 1 + 10560 T + p^{4} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12480 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1918 T + p^{4} T^{2} )( 1 + 18720 T + p^{4} T^{2} ) \) |
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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
−13.89400578871956669787745412891, −13.69305318176123270065613367331, −13.26341086887068863433865053223, −13.24591340123248497948621124324, −12.27494608196340888981068538299, −11.40681754861243979244633076410, −10.67883306758921087979535834539, −10.20859878437868529184728152512, −9.843236600845202435878174244280, −9.048500562781009169302877488581, −8.488841851247608132595691894411, −8.482160409172662208637888244667, −6.63797475190271377903195889818, −6.44775544082419244039483329403, −5.69767710173230427540313018400, −5.18147606317297128160351779991, −4.17698786116898767948633062169, −3.14957382961274861342249962076, −1.85645821910307486912262125648, −1.09530642933255520699695991395,
1.09530642933255520699695991395, 1.85645821910307486912262125648, 3.14957382961274861342249962076, 4.17698786116898767948633062169, 5.18147606317297128160351779991, 5.69767710173230427540313018400, 6.44775544082419244039483329403, 6.63797475190271377903195889818, 8.482160409172662208637888244667, 8.488841851247608132595691894411, 9.048500562781009169302877488581, 9.843236600845202435878174244280, 10.20859878437868529184728152512, 10.67883306758921087979535834539, 11.40681754861243979244633076410, 12.27494608196340888981068538299, 13.24591340123248497948621124324, 13.26341086887068863433865053223, 13.69305318176123270065613367331, 13.89400578871956669787745412891
