| L(s) = 1 | − 2-s + 4-s − 3·5-s − 8-s + 3·10-s + 16-s + 19-s − 3·20-s − 3·23-s + 4·25-s − 3·29-s − 32-s − 38-s + 3·40-s + 19·43-s + 3·46-s − 12·47-s − 10·49-s − 4·50-s + 15·53-s + 3·58-s + 64-s − 17·67-s + 15·71-s + 4·73-s + 76-s − 3·80-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 1/4·16-s + 0.229·19-s − 0.670·20-s − 0.625·23-s + 4/5·25-s − 0.557·29-s − 0.176·32-s − 0.162·38-s + 0.474·40-s + 2.89·43-s + 0.442·46-s − 1.75·47-s − 1.42·49-s − 0.565·50-s + 2.06·53-s + 0.393·58-s + 1/8·64-s − 2.07·67-s + 1.78·71-s + 0.468·73-s + 0.114·76-s − 0.335·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.592024887147004238522478109400, −8.191195454658669617417835931001, −7.81293625892389359448959552865, −7.32063813036665890781783507491, −7.12734422063696952927030048463, −6.29167561017585931333796614613, −5.99135520826587030680866696404, −5.23616651406848519937671396223, −4.67460018755356754408633544587, −3.95085073190493848317073200703, −3.67797931095645385521420582580, −2.87648326950427116269956643477, −2.18190094213517380836854554891, −1.12540824385951466905642356085, 0,
1.12540824385951466905642356085, 2.18190094213517380836854554891, 2.87648326950427116269956643477, 3.67797931095645385521420582580, 3.95085073190493848317073200703, 4.67460018755356754408633544587, 5.23616651406848519937671396223, 5.99135520826587030680866696404, 6.29167561017585931333796614613, 7.12734422063696952927030048463, 7.32063813036665890781783507491, 7.81293625892389359448959552865, 8.191195454658669617417835931001, 8.592024887147004238522478109400
