| L(s) = 1 | + 4-s + 4·5-s − 6·9-s + 4·11-s + 16-s − 4·19-s + 4·20-s + 11·25-s + 4·29-s − 6·36-s + 12·41-s + 4·44-s − 24·45-s + 2·49-s + 16·55-s + 24·59-s − 16·61-s + 64-s − 4·76-s − 24·79-s + 4·80-s + 27·81-s − 12·89-s − 16·95-s − 24·99-s + 11·100-s − 20·101-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.78·5-s − 2·9-s + 1.20·11-s + 1/4·16-s − 0.917·19-s + 0.894·20-s + 11/5·25-s + 0.742·29-s − 36-s + 1.87·41-s + 0.603·44-s − 3.57·45-s + 2/7·49-s + 2.15·55-s + 3.12·59-s − 2.04·61-s + 1/8·64-s − 0.458·76-s − 2.70·79-s + 0.447·80-s + 3·81-s − 1.27·89-s − 1.64·95-s − 2.41·99-s + 1.09·100-s − 1.99·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.953403245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.953403245\) |
| \(L(\frac{3}{2})\) |
|
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.928280836928168979628777920327, −9.594811427439617650906242640583, −8.971935452602447588346290189052, −8.651590213691851109739649270908, −8.276543750439528714060318536349, −7.29468472679713756407763618174, −6.69441913641732873591204403077, −6.21475081018625957585151869260, −5.84432417959727908710252168629, −5.50098265982593501879266203197, −4.63556271802396925268184937965, −3.74894127876684642815312676386, −2.63232737474552247848840602025, −2.53491101319297313540699620659, −1.35564478109443112808861515260,
1.35564478109443112808861515260, 2.53491101319297313540699620659, 2.63232737474552247848840602025, 3.74894127876684642815312676386, 4.63556271802396925268184937965, 5.50098265982593501879266203197, 5.84432417959727908710252168629, 6.21475081018625957585151869260, 6.69441913641732873591204403077, 7.29468472679713756407763618174, 8.276543750439528714060318536349, 8.651590213691851109739649270908, 8.971935452602447588346290189052, 9.594811427439617650906242640583, 9.928280836928168979628777920327
