| L(s) = 1 | + 2-s + 4-s + 8-s − 6·9-s + 16-s − 12·17-s − 6·18-s + 8·19-s − 6·25-s + 32-s − 12·34-s − 6·36-s + 8·38-s − 12·41-s + 16·43-s − 14·49-s − 6·50-s − 24·59-s + 64-s − 24·67-s − 12·68-s − 6·72-s + 20·73-s + 8·76-s + 27·81-s − 12·82-s + 16·83-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2·9-s + 1/4·16-s − 2.91·17-s − 1.41·18-s + 1.83·19-s − 6/5·25-s + 0.176·32-s − 2.05·34-s − 36-s + 1.29·38-s − 1.87·41-s + 2.43·43-s − 2·49-s − 0.848·50-s − 3.12·59-s + 1/8·64-s − 2.93·67-s − 1.45·68-s − 0.707·72-s + 2.34·73-s + 0.917·76-s + 3·81-s − 1.32·82-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.056498267458590117851162621111, −8.911717137174663405595486729622, −8.067419021434777137162075637049, −7.76857292337562945056886541715, −7.17663415905550465853772545662, −6.32858579420662158347624019611, −6.23927644667191941192015479418, −5.67007372238293440407484071620, −4.85213555842603099342069383801, −4.74203912193695697715499891246, −3.73552653057987685997850390510, −3.16188554299090604324575557253, −2.58717477143022626568369362485, −1.85194090492113304245340785765, 0,
1.85194090492113304245340785765, 2.58717477143022626568369362485, 3.16188554299090604324575557253, 3.73552653057987685997850390510, 4.74203912193695697715499891246, 4.85213555842603099342069383801, 5.67007372238293440407484071620, 6.23927644667191941192015479418, 6.32858579420662158347624019611, 7.17663415905550465853772545662, 7.76857292337562945056886541715, 8.067419021434777137162075637049, 8.911717137174663405595486729622, 9.056498267458590117851162621111
