| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s + 4·7-s + 4·8-s − 9-s + 4·10-s + 4·11-s − 6·12-s + 8·14-s − 4·15-s + 5·16-s − 2·17-s − 2·18-s − 6·19-s + 6·20-s − 8·21-s + 8·22-s − 8·24-s + 3·25-s + 6·27-s + 12·28-s + 4·29-s − 8·30-s + 16·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 1.51·7-s + 1.41·8-s − 1/3·9-s + 1.26·10-s + 1.20·11-s − 1.73·12-s + 2.13·14-s − 1.03·15-s + 5/4·16-s − 0.485·17-s − 0.471·18-s − 1.37·19-s + 1.34·20-s − 1.74·21-s + 1.70·22-s − 1.63·24-s + 3/5·25-s + 1.15·27-s + 2.26·28-s + 0.742·29-s − 1.46·30-s + 2.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.255423283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.255423283\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 16 T + 4 p T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 77 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 112 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 168 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 47 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 121 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 192 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 93 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.173036857877128173009720136069, −8.118405136000385591531591137512, −7.56411193155271772586968703833, −6.98882262869816183711404183654, −6.66713543333082081757058061389, −6.29433168583176606750276530379, −6.14640332262436909893387406197, −6.01162108020513210339172361452, −5.40182339648774373638705238830, −5.05123867212245688029291574403, −4.70020923990107444268882197545, −4.47264338901717540496896140140, −4.29939478879532896076085707802, −3.59037044088172993961796358026, −3.06956092966487623102972122029, −2.62045063049015991091179663335, −1.98164228744792650811789619903, −1.98128238251250156519790483403, −1.06557294274954510422702968714, −0.76563286882215492044374834908,
0.76563286882215492044374834908, 1.06557294274954510422702968714, 1.98128238251250156519790483403, 1.98164228744792650811789619903, 2.62045063049015991091179663335, 3.06956092966487623102972122029, 3.59037044088172993961796358026, 4.29939478879532896076085707802, 4.47264338901717540496896140140, 4.70020923990107444268882197545, 5.05123867212245688029291574403, 5.40182339648774373638705238830, 6.01162108020513210339172361452, 6.14640332262436909893387406197, 6.29433168583176606750276530379, 6.66713543333082081757058061389, 6.98882262869816183711404183654, 7.56411193155271772586968703833, 8.118405136000385591531591137512, 8.173036857877128173009720136069
