| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·5-s − 4·6-s + 4·8-s + 3·9-s − 8·10-s − 6·12-s + 8·15-s + 5·16-s + 6·18-s + 4·19-s − 12·20-s − 2·23-s − 8·24-s + 2·25-s − 4·27-s − 4·29-s + 16·30-s − 4·31-s + 6·32-s + 9·36-s − 4·37-s + 8·38-s − 16·40-s + 12·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.78·5-s − 1.63·6-s + 1.41·8-s + 9-s − 2.52·10-s − 1.73·12-s + 2.06·15-s + 5/4·16-s + 1.41·18-s + 0.917·19-s − 2.68·20-s − 0.417·23-s − 1.63·24-s + 2/5·25-s − 0.769·27-s − 0.742·29-s + 2.92·30-s − 0.718·31-s + 1.06·32-s + 3/2·36-s − 0.657·37-s + 1.29·38-s − 2.52·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 14 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.56949522208829419460220157969, −7.46240187672602840361923130008, −7.12384169358857544224533234657, −6.68561576903558979322592974762, −6.23024237043829594777686184890, −6.00086531050659593905619193889, −5.55498928191957357784180483372, −5.38527935029534650080003363422, −4.90097811643310037092845697735, −4.51389790047837359627473796013, −4.18153104469561100531937631122, −3.94143220501659481170953589650, −3.43642960235211622329161557337, −3.43440665376120551085979052233, −2.45568744021517076318742775169, −2.39710679754217439154470131666, −1.29491499148409965249544355272, −1.26458621099679501445844531273, 0, 0,
1.26458621099679501445844531273, 1.29491499148409965249544355272, 2.39710679754217439154470131666, 2.45568744021517076318742775169, 3.43440665376120551085979052233, 3.43642960235211622329161557337, 3.94143220501659481170953589650, 4.18153104469561100531937631122, 4.51389790047837359627473796013, 4.90097811643310037092845697735, 5.38527935029534650080003363422, 5.55498928191957357784180483372, 6.00086531050659593905619193889, 6.23024237043829594777686184890, 6.68561576903558979322592974762, 7.12384169358857544224533234657, 7.46240187672602840361923130008, 7.56949522208829419460220157969
