| L(s) = 1 | + 2·3-s + 2·4-s − 3·5-s + 9-s + 4·12-s − 6·15-s − 6·20-s + 9·23-s + 4·25-s − 4·27-s + 2·31-s + 2·36-s + 2·37-s − 3·45-s − 3·47-s − 2·49-s + 6·53-s − 15·59-s − 12·60-s − 8·64-s − 4·67-s + 18·69-s − 21·71-s + 8·75-s − 11·81-s + 9·89-s + 18·92-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 4-s − 1.34·5-s + 1/3·9-s + 1.15·12-s − 1.54·15-s − 1.34·20-s + 1.87·23-s + 4/5·25-s − 0.769·27-s + 0.359·31-s + 1/3·36-s + 0.328·37-s − 0.447·45-s − 0.437·47-s − 2/7·49-s + 0.824·53-s − 1.95·59-s − 1.54·60-s − 64-s − 0.488·67-s + 2.16·69-s − 2.49·71-s + 0.923·75-s − 1.22·81-s + 0.953·89-s + 1.87·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 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
−7.34756777576650051168982575516, −7.09151703389172071570779375970, −6.69262788913065898083037154346, −6.06793432993445778961105442807, −5.85264403529183768793985571586, −4.95142067594325456354149414745, −4.72474810492383340680468051662, −4.21634777490546200581758944721, −3.62745758228872065953096497687, −3.24553198791295731657302119183, −2.85876087849156628093692285308, −2.50720547629111837091247746292, −1.76876553601419242309310801336, −1.12894237257338866341798982582, 0,
1.12894237257338866341798982582, 1.76876553601419242309310801336, 2.50720547629111837091247746292, 2.85876087849156628093692285308, 3.24553198791295731657302119183, 3.62745758228872065953096497687, 4.21634777490546200581758944721, 4.72474810492383340680468051662, 4.95142067594325456354149414745, 5.85264403529183768793985571586, 6.06793432993445778961105442807, 6.69262788913065898083037154346, 7.09151703389172071570779375970, 7.34756777576650051168982575516
