| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s − 8·13-s + 16-s − 12·17-s − 18-s − 10·25-s + 8·26-s + 12·29-s − 32-s + 12·34-s + 36-s − 20·37-s + 12·41-s − 10·49-s + 10·50-s − 8·52-s − 12·58-s + 16·61-s + 64-s − 12·68-s − 72-s + 4·73-s + 20·74-s + 81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1/3·9-s − 2.21·13-s + 1/4·16-s − 2.91·17-s − 0.235·18-s − 2·25-s + 1.56·26-s + 2.22·29-s − 0.176·32-s + 2.05·34-s + 1/6·36-s − 3.28·37-s + 1.87·41-s − 1.42·49-s + 1.41·50-s − 1.10·52-s − 1.57·58-s + 2.04·61-s + 1/8·64-s − 1.45·68-s − 0.117·72-s + 0.468·73-s + 2.32·74-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−10.21971960261641202232640310225, −9.702896608623430445687767205931, −8.958855021658912251315559727512, −8.767992898257258380407868928000, −8.013104010237496671597514230187, −7.47999694363717860882978383906, −6.77506565559004391547494912024, −6.73373984274544568686272664379, −5.80771575321423739175967870457, −4.75034201814524111107553764909, −4.69333471332706230755932945396, −3.62485072881439717663109857391, −2.37545091357255938865951638025, −2.12574870769826326935834056810, 0,
2.12574870769826326935834056810, 2.37545091357255938865951638025, 3.62485072881439717663109857391, 4.69333471332706230755932945396, 4.75034201814524111107553764909, 5.80771575321423739175967870457, 6.73373984274544568686272664379, 6.77506565559004391547494912024, 7.47999694363717860882978383906, 8.013104010237496671597514230187, 8.767992898257258380407868928000, 8.958855021658912251315559727512, 9.702896608623430445687767205931, 10.21971960261641202232640310225
