| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s − 4·10-s + 6·12-s − 6·13-s − 4·14-s + 4·15-s + 5·16-s + 10·17-s − 6·18-s + 2·19-s + 6·20-s + 4·21-s − 2·23-s − 8·24-s + 3·25-s + 12·26-s + 4·27-s + 6·28-s − 8·30-s + 2·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s + 1.73·12-s − 1.66·13-s − 1.06·14-s + 1.03·15-s + 5/4·16-s + 2.42·17-s − 1.41·18-s + 0.458·19-s + 1.34·20-s + 0.872·21-s − 0.417·23-s − 1.63·24-s + 3/5·25-s + 2.35·26-s + 0.769·27-s + 1.13·28-s − 1.46·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.145743075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.145743075\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 146 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T - 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 214 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.318396078945771702797837419086, −8.109582827754720537081000194400, −7.74686742140539534415989933015, −7.63669471938399930903709446897, −7.19171249660702718705469167131, −7.05121535907198423251562008417, −6.20858652601148860934792952675, −6.16234544237514628715073382798, −5.52169069006278871877121246092, −5.29441372713179410011441410125, −4.77202134039756339987543860361, −4.45170267606560348673336192998, −3.61757211642494421972547230429, −3.44959302005306987676329316914, −2.72134924713032012478403362952, −2.60810795492820535960390202053, −2.04960736964114162359825104117, −1.74685429925845420887555756229, −1.00432686044453438847419069030, −0.76988346468970602930976530904,
0.76988346468970602930976530904, 1.00432686044453438847419069030, 1.74685429925845420887555756229, 2.04960736964114162359825104117, 2.60810795492820535960390202053, 2.72134924713032012478403362952, 3.44959302005306987676329316914, 3.61757211642494421972547230429, 4.45170267606560348673336192998, 4.77202134039756339987543860361, 5.29441372713179410011441410125, 5.52169069006278871877121246092, 6.16234544237514628715073382798, 6.20858652601148860934792952675, 7.05121535907198423251562008417, 7.19171249660702718705469167131, 7.63669471938399930903709446897, 7.74686742140539534415989933015, 8.109582827754720537081000194400, 8.318396078945771702797837419086
