| L(s) = 1 | − 4-s − 2·9-s − 3·16-s − 2·19-s − 5·25-s + 10·31-s + 2·36-s + 18·41-s − 4·49-s − 6·59-s − 14·61-s + 7·64-s + 5·71-s + 2·76-s + 10·79-s − 5·81-s + 24·89-s + 5·100-s − 12·101-s − 20·109-s − 22·121-s − 10·124-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2/3·9-s − 3/4·16-s − 0.458·19-s − 25-s + 1.79·31-s + 1/3·36-s + 2.81·41-s − 4/7·49-s − 0.781·59-s − 1.79·61-s + 7/8·64-s + 0.593·71-s + 0.229·76-s + 1.12·79-s − 5/9·81-s + 2.54·89-s + 1/2·100-s − 1.19·101-s − 1.91·109-s − 2·121-s − 0.898·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/2·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5562389841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5562389841\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−13.60087858255578935187610056587, −12.93187818647123957581627206874, −12.21268381036832588509940488243, −11.65596610109153528441266581074, −10.97884074586101149554059695463, −10.39416229328768790005077075621, −9.371334827491565086249240247667, −9.187311544451760090050200437835, −8.172056324467741840973684754598, −7.72725555759426172762942762129, −6.53311959388175250320155123004, −5.94849071130610912286020064781, −4.85622802330129372995173241780, −4.06479377195573508697148007868, −2.63275119095852124695005682804,
2.63275119095852124695005682804, 4.06479377195573508697148007868, 4.85622802330129372995173241780, 5.94849071130610912286020064781, 6.53311959388175250320155123004, 7.72725555759426172762942762129, 8.172056324467741840973684754598, 9.187311544451760090050200437835, 9.371334827491565086249240247667, 10.39416229328768790005077075621, 10.97884074586101149554059695463, 11.65596610109153528441266581074, 12.21268381036832588509940488243, 12.93187818647123957581627206874, 13.60087858255578935187610056587
