| L(s) = 1 | + 3-s + 4-s − 2·9-s + 5·11-s + 12-s − 3·16-s − 5·23-s − 5·27-s + 5·31-s + 5·33-s − 2·36-s − 9·37-s + 5·44-s − 5·47-s − 3·48-s − 11·49-s − 20·59-s − 7·64-s − 16·67-s − 5·69-s − 10·71-s + 81-s + 15·89-s − 5·92-s + 5·93-s + 9·97-s − 10·99-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 2/3·9-s + 1.50·11-s + 0.288·12-s − 3/4·16-s − 1.04·23-s − 0.962·27-s + 0.898·31-s + 0.870·33-s − 1/3·36-s − 1.47·37-s + 0.753·44-s − 0.729·47-s − 0.433·48-s − 1.57·49-s − 2.60·59-s − 7/8·64-s − 1.95·67-s − 0.601·69-s − 1.18·71-s + 1/9·81-s + 1.58·89-s − 0.521·92-s + 0.518·93-s + 0.913·97-s − 1.00·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.067968704912731268715246186950, −7.70629984332701510511846414360, −7.31918762549819813856891142784, −6.56953429526110025820310810710, −6.37943766312733801780826663617, −6.10713077897795158043766563778, −5.38491845367795989335977168293, −4.71699318681551725304329690997, −4.34898679927011385468749545500, −3.68621614289464050729887562005, −3.18967823488865577608601639758, −2.74413397281607218087577881660, −1.84454934666866023488186434514, −1.54968263648409039517177388483, 0,
1.54968263648409039517177388483, 1.84454934666866023488186434514, 2.74413397281607218087577881660, 3.18967823488865577608601639758, 3.68621614289464050729887562005, 4.34898679927011385468749545500, 4.71699318681551725304329690997, 5.38491845367795989335977168293, 6.10713077897795158043766563778, 6.37943766312733801780826663617, 6.56953429526110025820310810710, 7.31918762549819813856891142784, 7.70629984332701510511846414360, 8.067968704912731268715246186950
