| L(s) = 1 | − 3-s + 2·5-s + 9-s + 4·13-s − 2·15-s + 4·17-s − 8·19-s − 25-s − 27-s − 12·29-s − 12·37-s − 4·39-s + 2·45-s − 14·49-s − 4·51-s + 8·57-s + 8·65-s − 16·71-s + 75-s + 81-s − 8·83-s + 8·85-s + 12·87-s − 16·95-s + 36·101-s − 32·103-s − 24·107-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.10·13-s − 0.516·15-s + 0.970·17-s − 1.83·19-s − 1/5·25-s − 0.192·27-s − 2.22·29-s − 1.97·37-s − 0.640·39-s + 0.298·45-s − 2·49-s − 0.560·51-s + 1.05·57-s + 0.992·65-s − 1.89·71-s + 0.115·75-s + 1/9·81-s − 0.878·83-s + 0.867·85-s + 1.28·87-s − 1.64·95-s + 3.58·101-s − 3.15·103-s − 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.730633394111998328725718228031, −8.095549656698690551636863836910, −7.47308398754126662115910632702, −7.13146007277855219299925953915, −6.33548194788761977078999997603, −6.21040581024747445781537308376, −5.74245729062321480161908713186, −5.24179959934618595033608799542, −4.76184067616314722575220108003, −3.84771560491769769437106298900, −3.71523857372513194753595622108, −2.79864903411763483594224581992, −1.72979115909686153918413689995, −1.64229196717434907696557687175, 0,
1.64229196717434907696557687175, 1.72979115909686153918413689995, 2.79864903411763483594224581992, 3.71523857372513194753595622108, 3.84771560491769769437106298900, 4.76184067616314722575220108003, 5.24179959934618595033608799542, 5.74245729062321480161908713186, 6.21040581024747445781537308376, 6.33548194788761977078999997603, 7.13146007277855219299925953915, 7.47308398754126662115910632702, 8.095549656698690551636863836910, 8.730633394111998328725718228031
