| L(s) = 1 | + 2-s + 4-s + 8-s + 2·9-s + 8·11-s + 16-s + 2·18-s + 8·22-s + 25-s + 32-s + 2·36-s − 8·43-s + 8·44-s − 7·49-s + 50-s + 64-s − 8·67-s + 2·72-s − 5·81-s − 8·86-s + 8·88-s − 7·98-s + 16·99-s + 100-s − 24·107-s + 4·113-s + 26·121-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2/3·9-s + 2.41·11-s + 1/4·16-s + 0.471·18-s + 1.70·22-s + 1/5·25-s + 0.176·32-s + 1/3·36-s − 1.21·43-s + 1.20·44-s − 49-s + 0.141·50-s + 1/8·64-s − 0.977·67-s + 0.235·72-s − 5/9·81-s − 0.862·86-s + 0.852·88-s − 0.707·98-s + 1.60·99-s + 1/10·100-s − 2.32·107-s + 0.376·113-s + 2.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.181205060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.181205060\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−9.228681351790577326469561001469, −8.889273761166958937026182222853, −8.286613435008629958738844034419, −7.72328747083391192454565538650, −7.08186843693279573059730100827, −6.68913565368302242484324187620, −6.42890995300149693126761555030, −5.82088745937099503373811770260, −5.13389681027056304099492968073, −4.50374688279707185739116659424, −4.08403578752006591140007099921, −3.58742867664654545697808399121, −2.92820897979558166724714199170, −1.80282640489664796934885953887, −1.28157127838213072824799933994,
1.28157127838213072824799933994, 1.80282640489664796934885953887, 2.92820897979558166724714199170, 3.58742867664654545697808399121, 4.08403578752006591140007099921, 4.50374688279707185739116659424, 5.13389681027056304099492968073, 5.82088745937099503373811770260, 6.42890995300149693126761555030, 6.68913565368302242484324187620, 7.08186843693279573059730100827, 7.72328747083391192454565538650, 8.286613435008629958738844034419, 8.889273761166958937026182222853, 9.228681351790577326469561001469
