| L(s) = 1 | + 2·2-s + 4-s + 4·7-s − 2·11-s + 8·13-s + 8·14-s + 16-s + 8·17-s − 4·22-s + 16·26-s + 4·28-s − 4·29-s − 2·32-s + 16·34-s + 4·37-s − 12·41-s + 12·43-s − 2·44-s − 2·49-s + 8·52-s + 12·53-s − 8·58-s + 8·59-s + 4·61-s − 11·64-s − 8·67-s + 8·68-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 1.51·7-s − 0.603·11-s + 2.21·13-s + 2.13·14-s + 1/4·16-s + 1.94·17-s − 0.852·22-s + 3.13·26-s + 0.755·28-s − 0.742·29-s − 0.353·32-s + 2.74·34-s + 0.657·37-s − 1.87·41-s + 1.82·43-s − 0.301·44-s − 2/7·49-s + 1.10·52-s + 1.64·53-s − 1.05·58-s + 1.04·59-s + 0.512·61-s − 1.37·64-s − 0.977·67-s + 0.970·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.275530146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.275530146\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.775792573652341552029370576849, −8.756526380290448756108932598459, −8.348195479592099039830851439412, −7.84436201939499780430693494726, −7.66396827459484780518990616215, −7.32837030396739543953959643528, −6.66873596362348492184392959843, −6.19136387220385288591921419836, −5.71026623909183260703198512596, −5.58917887616144134891618195671, −5.10947350956089652792015005920, −4.92272567370070635036782669109, −4.23103054301320907439201112597, −4.01689957358453246539912196379, −3.45101516937177529512010353420, −3.36029176631802819053516109770, −2.52939341252480033016628415450, −1.84988382650529938759860759880, −1.35717830261851675732069691262, −0.859833060999134482874499174925,
0.859833060999134482874499174925, 1.35717830261851675732069691262, 1.84988382650529938759860759880, 2.52939341252480033016628415450, 3.36029176631802819053516109770, 3.45101516937177529512010353420, 4.01689957358453246539912196379, 4.23103054301320907439201112597, 4.92272567370070635036782669109, 5.10947350956089652792015005920, 5.58917887616144134891618195671, 5.71026623909183260703198512596, 6.19136387220385288591921419836, 6.66873596362348492184392959843, 7.32837030396739543953959643528, 7.66396827459484780518990616215, 7.84436201939499780430693494726, 8.348195479592099039830851439412, 8.756526380290448756108932598459, 8.775792573652341552029370576849
