| L(s) = 1 | + 3-s + 4-s − 2·5-s + 7-s + 9-s + 12-s − 2·15-s + 16-s − 12·17-s − 2·20-s + 21-s + 3·25-s + 27-s + 28-s − 2·35-s + 36-s + 4·37-s + 12·41-s + 16·43-s − 2·45-s − 24·47-s + 48-s + 49-s − 12·51-s − 24·59-s − 2·60-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.288·12-s − 0.516·15-s + 1/4·16-s − 2.91·17-s − 0.447·20-s + 0.218·21-s + 3/5·25-s + 0.192·27-s + 0.188·28-s − 0.338·35-s + 1/6·36-s + 0.657·37-s + 1.87·41-s + 2.43·43-s − 0.298·45-s − 3.50·47-s + 0.144·48-s + 1/7·49-s − 1.68·51-s − 3.12·59-s − 0.258·60-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.991006296285731130234927884691, −7.41880334400769005077113607898, −7.27965583167809929942184453108, −6.66525158720960711652638814741, −6.14620005069006563789988203374, −6.03347639479763526485218429964, −4.88823096770031111535459012199, −4.64849576847169223614699526425, −4.31977202761190735797982138368, −3.71822253537069739639390686069, −3.12590542097473718225449308769, −2.43310152932511024940703619852, −2.16105217917237072156284139202, −1.22061090343879583892735969324, 0,
1.22061090343879583892735969324, 2.16105217917237072156284139202, 2.43310152932511024940703619852, 3.12590542097473718225449308769, 3.71822253537069739639390686069, 4.31977202761190735797982138368, 4.64849576847169223614699526425, 4.88823096770031111535459012199, 6.03347639479763526485218429964, 6.14620005069006563789988203374, 6.66525158720960711652638814741, 7.27965583167809929942184453108, 7.41880334400769005077113607898, 7.991006296285731130234927884691
