| L(s) = 1 | − 3-s + 4-s + 4·7-s + 9-s − 12-s − 2·13-s + 16-s − 12·19-s − 4·21-s + 25-s − 27-s + 4·28-s + 36-s − 4·37-s + 2·39-s − 8·43-s − 48-s − 2·49-s − 2·52-s + 12·57-s − 20·61-s + 4·63-s + 64-s + 24·67-s − 16·73-s − 75-s − 12·76-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 1.51·7-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 2.75·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s + 0.755·28-s + 1/6·36-s − 0.657·37-s + 0.320·39-s − 1.21·43-s − 0.144·48-s − 2/7·49-s − 0.277·52-s + 1.58·57-s − 2.56·61-s + 0.503·63-s + 1/8·64-s + 2.93·67-s − 1.87·73-s − 0.115·75-s − 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−8.380996086562014846279025757970, −8.020484731538067335126892167403, −7.38229162108116731403430084124, −6.91405802728647033780426052154, −6.46010665238686597416245482811, −6.17877302783647145193418517915, −5.36890034019942549244695495877, −5.08920925224039309093622955852, −4.46780194170664996451904659653, −4.24180232783324699757717994198, −3.39888206557913870378261957815, −2.53385850326612349329092506900, −1.92741690745051271953551936931, −1.48277331421431160180933642843, 0,
1.48277331421431160180933642843, 1.92741690745051271953551936931, 2.53385850326612349329092506900, 3.39888206557913870378261957815, 4.24180232783324699757717994198, 4.46780194170664996451904659653, 5.08920925224039309093622955852, 5.36890034019942549244695495877, 6.17877302783647145193418517915, 6.46010665238686597416245482811, 6.91405802728647033780426052154, 7.38229162108116731403430084124, 8.020484731538067335126892167403, 8.380996086562014846279025757970
