| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s + 4·9-s + 6·11-s + 4·14-s + 16-s − 4·18-s − 6·22-s − 23-s − 4·25-s − 4·28-s + 6·29-s − 32-s + 4·36-s + 4·37-s + 4·43-s + 6·44-s + 46-s + 9·49-s + 4·50-s − 6·53-s + 4·56-s − 6·58-s − 16·63-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 4/3·9-s + 1.80·11-s + 1.06·14-s + 1/4·16-s − 0.942·18-s − 1.27·22-s − 0.208·23-s − 4/5·25-s − 0.755·28-s + 1.11·29-s − 0.176·32-s + 2/3·36-s + 0.657·37-s + 0.609·43-s + 0.904·44-s + 0.147·46-s + 9/7·49-s + 0.565·50-s − 0.824·53-s + 0.534·56-s − 0.787·58-s − 2.01·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7237516486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7237516486\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
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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
−11.77826397849150309561313347605, −10.94002108150049721864584251057, −10.24079750317978586112492297642, −9.901731715378522654840576692729, −9.356226868287216479824902337231, −9.071304304422187122372023939472, −8.227982389244717579397016529301, −7.35231816043881306251859575892, −6.95981812449191239444473645750, −6.31033131434401419946857176520, −5.95649740093807782541197607009, −4.44716647603505583147667839235, −3.91435098179270223593045060919, −2.94864473125324080516458733393, −1.44068337440778745531144755060,
1.44068337440778745531144755060, 2.94864473125324080516458733393, 3.91435098179270223593045060919, 4.44716647603505583147667839235, 5.95649740093807782541197607009, 6.31033131434401419946857176520, 6.95981812449191239444473645750, 7.35231816043881306251859575892, 8.227982389244717579397016529301, 9.071304304422187122372023939472, 9.356226868287216479824902337231, 9.901731715378522654840576692729, 10.24079750317978586112492297642, 10.94002108150049721864584251057, 11.77826397849150309561313347605
