| L(s) = 1 | + 2·2-s + 3·4-s − 5-s + 2·7-s + 4·8-s − 2·10-s + 8·11-s + 5·13-s + 4·14-s + 5·16-s − 8·17-s + 2·19-s − 3·20-s + 16·22-s − 2·23-s + 25-s + 10·26-s + 6·28-s − 7·29-s − 4·31-s + 6·32-s − 16·34-s − 2·35-s + 15·37-s + 4·38-s − 4·40-s + 7·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.447·5-s + 0.755·7-s + 1.41·8-s − 0.632·10-s + 2.41·11-s + 1.38·13-s + 1.06·14-s + 5/4·16-s − 1.94·17-s + 0.458·19-s − 0.670·20-s + 3.41·22-s − 0.417·23-s + 1/5·25-s + 1.96·26-s + 1.13·28-s − 1.29·29-s − 0.718·31-s + 1.06·32-s − 2.74·34-s − 0.338·35-s + 2.46·37-s + 0.648·38-s − 0.632·40-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.06189890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.06189890\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 15 T + 120 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 78 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 162 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 19 T + 274 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
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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
−9.126757857835755877839062312135, −8.500622646010457550698242244657, −8.146652912397608881474640982753, −7.77559038018779645695992737940, −7.17492355305916466483826233031, −7.05225369316589681007727539576, −6.59469215707706262623068516039, −6.14073453144146507667082244266, −5.86800538294389149040174988243, −5.71794245741845818685227151327, −4.87896623063302869473157119088, −4.48266346907206762502895641965, −4.15045559389074793520254337776, −4.00730420632544595064044240466, −3.59550586560046086395018239392, −3.10296351108977347968222332467, −2.20284918099536179505614135847, −2.09490712769327407997606593046, −1.29393395565374479541145938594, −0.882849108044602374910707935437,
0.882849108044602374910707935437, 1.29393395565374479541145938594, 2.09490712769327407997606593046, 2.20284918099536179505614135847, 3.10296351108977347968222332467, 3.59550586560046086395018239392, 4.00730420632544595064044240466, 4.15045559389074793520254337776, 4.48266346907206762502895641965, 4.87896623063302869473157119088, 5.71794245741845818685227151327, 5.86800538294389149040174988243, 6.14073453144146507667082244266, 6.59469215707706262623068516039, 7.05225369316589681007727539576, 7.17492355305916466483826233031, 7.77559038018779645695992737940, 8.146652912397608881474640982753, 8.500622646010457550698242244657, 9.126757857835755877839062312135
