| L(s) = 1 | + 2-s + 4-s + 8-s + 3·9-s − 12·13-s + 16-s − 2·17-s + 3·18-s − 12·26-s − 12·29-s + 32-s − 2·34-s + 3·36-s + 16·37-s + 22·41-s + 49-s − 12·52-s + 8·53-s − 12·58-s − 4·61-s + 64-s − 2·68-s + 3·72-s − 14·73-s + 16·74-s + 22·82-s − 22·89-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 9-s − 3.32·13-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 2.35·26-s − 2.22·29-s + 0.176·32-s − 0.342·34-s + 1/2·36-s + 2.63·37-s + 3.43·41-s + 1/7·49-s − 1.66·52-s + 1.09·53-s − 1.57·58-s − 0.512·61-s + 1/8·64-s − 0.242·68-s + 0.353·72-s − 1.63·73-s + 1.85·74-s + 2.42·82-s − 2.33·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.70054391482804408966188304160, −7.48255673861422805516705752641, −6.95538626651508714624591948210, −6.88567847635591555925756251512, −5.84573096199937208498827543184, −5.73190365453661485063705700568, −5.20618442524944639616146822933, −4.45614212379368613554123897305, −4.32124854619714589271574708665, −4.07830139628673154988365646824, −2.95018645973909036381834936400, −2.44905781159727834481891285350, −2.30364756815994812074860517182, −1.25902608111017062377912870000, 0,
1.25902608111017062377912870000, 2.30364756815994812074860517182, 2.44905781159727834481891285350, 2.95018645973909036381834936400, 4.07830139628673154988365646824, 4.32124854619714589271574708665, 4.45614212379368613554123897305, 5.20618442524944639616146822933, 5.73190365453661485063705700568, 5.84573096199937208498827543184, 6.88567847635591555925756251512, 6.95538626651508714624591948210, 7.48255673861422805516705752641, 7.70054391482804408966188304160
