| L(s) = 1 | − 4·4-s − 5·9-s + 6·11-s + 12·16-s + 4·19-s − 5·25-s − 12·29-s − 8·31-s + 20·36-s − 18·41-s − 24·44-s − 13·49-s + 24·59-s + 16·61-s − 32·64-s − 30·71-s − 16·76-s − 20·79-s + 16·81-s + 12·89-s − 30·99-s + 20·100-s + 6·101-s + 4·109-s + 48·116-s + 5·121-s + 32·124-s + ⋯ |
| L(s) = 1 | − 2·4-s − 5/3·9-s + 1.80·11-s + 3·16-s + 0.917·19-s − 25-s − 2.22·29-s − 1.43·31-s + 10/3·36-s − 2.81·41-s − 3.61·44-s − 1.85·49-s + 3.12·59-s + 2.04·61-s − 4·64-s − 3.56·71-s − 1.83·76-s − 2.25·79-s + 16/9·81-s + 1.27·89-s − 3.01·99-s + 2·100-s + 0.597·101-s + 0.383·109-s + 4.45·116-s + 5/11·121-s + 2.87·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_2$ | \( 1 + p T^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.819966817497618882768827794755, −9.591268537738103586397612813688, −9.032345851694468357832029856913, −8.647494221806725911176641352691, −8.390237773813478013059373004459, −7.59911177067371416247669368567, −6.96328856044475469810412834337, −6.03232690048232103384500782557, −5.44973416215471530225317007593, −5.31354420552212102801103714157, −4.28699193481010687037956905954, −3.50910294340479626549928321873, −3.49397830742570960606550504227, −1.63648598905395091233402759288, 0,
1.63648598905395091233402759288, 3.49397830742570960606550504227, 3.50910294340479626549928321873, 4.28699193481010687037956905954, 5.31354420552212102801103714157, 5.44973416215471530225317007593, 6.03232690048232103384500782557, 6.96328856044475469810412834337, 7.59911177067371416247669368567, 8.390237773813478013059373004459, 8.647494221806725911176641352691, 9.032345851694468357832029856913, 9.591268537738103586397612813688, 9.819966817497618882768827794755
