| L(s) = 1 | + 2-s + 3-s − 4-s + 2·5-s + 6-s − 2·7-s − 3·8-s + 9-s + 2·10-s − 8·11-s − 12-s − 2·14-s + 2·15-s − 16-s + 12·17-s + 18-s − 2·20-s − 2·21-s − 8·22-s − 3·24-s − 25-s + 27-s + 2·28-s + 2·30-s + 5·32-s − 8·33-s + 12·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 2.41·11-s − 0.288·12-s − 0.534·14-s + 0.516·15-s − 1/4·16-s + 2.91·17-s + 0.235·18-s − 0.447·20-s − 0.436·21-s − 1.70·22-s − 0.612·24-s − 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.365·30-s + 0.883·32-s − 1.39·33-s + 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.098208619051863702360994501809, −7.81295430367747747098376616892, −7.61822574925806319174158632173, −6.83504788729234993876002700605, −6.04959426637495665136321555363, −5.94607665445287604157300008429, −5.39410355388125677723019647350, −5.02709416296405066539370067618, −4.63718162137583308032885665524, −3.66131632932358101038920576125, −3.15093821635969802098025587397, −3.05422074105458389777226041971, −2.31347905816165764850190874305, −1.35462327275534607341622822998, 0,
1.35462327275534607341622822998, 2.31347905816165764850190874305, 3.05422074105458389777226041971, 3.15093821635969802098025587397, 3.66131632932358101038920576125, 4.63718162137583308032885665524, 5.02709416296405066539370067618, 5.39410355388125677723019647350, 5.94607665445287604157300008429, 6.04959426637495665136321555363, 6.83504788729234993876002700605, 7.61822574925806319174158632173, 7.81295430367747747098376616892, 8.098208619051863702360994501809
