| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s − 14-s − 16-s + 6·25-s − 28-s − 5·32-s + 49-s − 6·50-s + 3·56-s + 7·64-s + 32·79-s − 98-s − 6·100-s − 112-s − 28·113-s − 6·121-s + 127-s + 3·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 0.267·14-s − 1/4·16-s + 6/5·25-s − 0.188·28-s − 0.883·32-s + 1/7·49-s − 0.848·50-s + 0.400·56-s + 7/8·64-s + 3.60·79-s − 0.101·98-s − 3/5·100-s − 0.0944·112-s − 2.63·113-s − 0.545·121-s + 0.0887·127-s + 0.265·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1778112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1778112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.228798671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.228798671\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.030514768084889787852043098059, −7.48733483740196396053226289227, −7.06594028263065496300867747391, −6.63478496161881034331199213050, −6.22628169315218824079820526079, −5.51186257816528055742974819242, −5.18431446031001017815063505262, −4.78405858770322399321481635281, −4.31510861551258387176284642505, −3.79891101220150690008710647647, −3.28648042377121578295375043576, −2.58732362847431751508943243155, −1.95450748716524399321738996754, −1.26168385839125974576745925039, −0.58503832335070089164870164959,
0.58503832335070089164870164959, 1.26168385839125974576745925039, 1.95450748716524399321738996754, 2.58732362847431751508943243155, 3.28648042377121578295375043576, 3.79891101220150690008710647647, 4.31510861551258387176284642505, 4.78405858770322399321481635281, 5.18431446031001017815063505262, 5.51186257816528055742974819242, 6.22628169315218824079820526079, 6.63478496161881034331199213050, 7.06594028263065496300867747391, 7.48733483740196396053226289227, 8.030514768084889787852043098059
