| L(s) = 1 | − 2-s + 4-s − 8-s + 6·11-s + 16-s + 6·19-s − 6·22-s + 4·25-s − 32-s − 6·38-s + 12·41-s − 4·43-s + 6·44-s + 6·49-s − 4·50-s − 6·59-s + 64-s − 10·67-s + 6·76-s − 12·82-s − 18·83-s + 4·86-s − 6·88-s + 24·89-s + 18·97-s − 6·98-s + 4·100-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.80·11-s + 1/4·16-s + 1.37·19-s − 1.27·22-s + 4/5·25-s − 0.176·32-s − 0.973·38-s + 1.87·41-s − 0.609·43-s + 0.904·44-s + 6/7·49-s − 0.565·50-s − 0.781·59-s + 1/8·64-s − 1.22·67-s + 0.688·76-s − 1.32·82-s − 1.97·83-s + 0.431·86-s − 0.639·88-s + 2.54·89-s + 1.82·97-s − 0.606·98-s + 2/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2996352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2996352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.242991812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.242991812\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
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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
−7.41319703822864624290069054477, −7.36679322197308847255299279982, −6.79017107993195925662348929503, −6.32999421058590206093011734317, −6.01067390828185076567126635771, −5.66207140416213785514804627143, −4.95094002050349790570173327307, −4.55975723488796393543347403105, −4.09108988102192877221591108994, −3.40022009220477496081588570383, −3.24006755681468727762726546468, −2.48592448897482691697940724855, −1.83744847796417281506613910031, −1.18897314202365691068725895052, −0.74982888546797453593009485552,
0.74982888546797453593009485552, 1.18897314202365691068725895052, 1.83744847796417281506613910031, 2.48592448897482691697940724855, 3.24006755681468727762726546468, 3.40022009220477496081588570383, 4.09108988102192877221591108994, 4.55975723488796393543347403105, 4.95094002050349790570173327307, 5.66207140416213785514804627143, 6.01067390828185076567126635771, 6.32999421058590206093011734317, 6.79017107993195925662348929503, 7.36679322197308847255299279982, 7.41319703822864624290069054477
