| L(s) = 1 | − 2·3-s + 9-s + 6·11-s − 6·23-s − 5·25-s + 4·27-s − 12·31-s − 12·33-s − 2·37-s − 6·47-s + 4·49-s + 10·53-s + 2·67-s + 12·69-s − 8·71-s + 10·75-s − 11·81-s + 12·89-s + 24·93-s + 10·97-s + 6·99-s + 14·103-s + 4·111-s + 6·113-s + 25·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.80·11-s − 1.25·23-s − 25-s + 0.769·27-s − 2.15·31-s − 2.08·33-s − 0.328·37-s − 0.875·47-s + 4/7·49-s + 1.37·53-s + 0.244·67-s + 1.44·69-s − 0.949·71-s + 1.15·75-s − 1.22·81-s + 1.27·89-s + 2.48·93-s + 1.01·97-s + 0.603·99-s + 1.37·103-s + 0.379·111-s + 0.564·113-s + 2.27·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−7.49210921191165194472594932445, −7.12859845530563254699686520903, −6.65449099208137376164286734092, −6.22759636719316512583037272183, −5.95749317697336908036998733640, −5.55182413151554932972084917920, −5.10085686126061416213536097625, −4.51535719774097932989144740548, −3.96201925891767710949017599817, −3.75083750582528864509509848682, −3.15271572467732655737810953170, −2.10532992933991790258524786101, −1.77955326635699234160861318624, −0.928139248007531457766375284407, 0,
0.928139248007531457766375284407, 1.77955326635699234160861318624, 2.10532992933991790258524786101, 3.15271572467732655737810953170, 3.75083750582528864509509848682, 3.96201925891767710949017599817, 4.51535719774097932989144740548, 5.10085686126061416213536097625, 5.55182413151554932972084917920, 5.95749317697336908036998733640, 6.22759636719316512583037272183, 6.65449099208137376164286734092, 7.12859845530563254699686520903, 7.49210921191165194472594932445
