| L(s) = 1 | + 2·2-s − 4·3-s + 3·4-s + 5-s − 8·6-s − 3·7-s + 4·8-s + 6·9-s + 2·10-s − 12·12-s − 4·13-s − 6·14-s − 4·15-s + 5·16-s + 9·17-s + 12·18-s + 2·19-s + 3·20-s + 12·21-s − 9·23-s − 16·24-s − 8·25-s − 8·26-s + 4·27-s − 9·28-s + 12·29-s − 8·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s + 3/2·4-s + 0.447·5-s − 3.26·6-s − 1.13·7-s + 1.41·8-s + 2·9-s + 0.632·10-s − 3.46·12-s − 1.10·13-s − 1.60·14-s − 1.03·15-s + 5/4·16-s + 2.18·17-s + 2.82·18-s + 0.458·19-s + 0.670·20-s + 2.61·21-s − 1.87·23-s − 3.26·24-s − 8/5·25-s − 1.56·26-s + 0.769·27-s − 1.70·28-s + 2.22·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 53 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 55 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 67 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 99 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 154 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 254 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 238 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78170163991278295346999971229, −7.65352371865183146561316864072, −7.25990791767006919086354467654, −6.61837419657795573612745364479, −6.34264032383615264793978943191, −6.23980455174058285061701918720, −5.91331107784492094247068093318, −5.66772664389623827503423585486, −5.24876326650737583986214761280, −4.90592583676155938046473541599, −4.62349981674817245075266384114, −4.25174262966563049299176686504, −3.44853846450588363611909517548, −3.35575911344486109989536598598, −2.68148181105250630854910565410, −2.51323547474406339157027294835, −1.35322078436533336793995264758, −1.30808828819426220615574303115, 0, 0,
1.30808828819426220615574303115, 1.35322078436533336793995264758, 2.51323547474406339157027294835, 2.68148181105250630854910565410, 3.35575911344486109989536598598, 3.44853846450588363611909517548, 4.25174262966563049299176686504, 4.62349981674817245075266384114, 4.90592583676155938046473541599, 5.24876326650737583986214761280, 5.66772664389623827503423585486, 5.91331107784492094247068093318, 6.23980455174058285061701918720, 6.34264032383615264793978943191, 6.61837419657795573612745364479, 7.25990791767006919086354467654, 7.65352371865183146561316864072, 7.78170163991278295346999971229
