| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s + 9-s − 2·10-s + 4·13-s + 16-s − 12·17-s + 18-s − 2·20-s + 3·25-s + 4·26-s − 12·29-s + 32-s − 12·34-s + 36-s + 4·37-s − 2·40-s + 12·41-s − 2·45-s + 49-s + 3·50-s + 4·52-s + 12·53-s − 12·58-s + 4·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.10·13-s + 1/4·16-s − 2.91·17-s + 0.235·18-s − 0.447·20-s + 3/5·25-s + 0.784·26-s − 2.22·29-s + 0.176·32-s − 2.05·34-s + 1/6·36-s + 0.657·37-s − 0.316·40-s + 1.87·41-s − 0.298·45-s + 1/7·49-s + 0.424·50-s + 0.554·52-s + 1.64·53-s − 1.57·58-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.181476201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.181476201\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−8.647026888430366294854058821975, −8.319627980737908137853866947983, −7.64537707540215722485320944275, −7.27965583167809929942184453108, −6.91673575142178820615272805521, −6.16116790751690085616531757265, −6.14620005069006563789988203374, −5.30537393915954566746162190631, −4.64849576847169223614699526425, −4.32773350578479900565119001608, −3.71822253537069739639390686069, −3.54574519606089903353001249150, −2.27238370722428728570015712288, −2.16105217917237072156284139202, −0.74026101682650381544781632886,
0.74026101682650381544781632886, 2.16105217917237072156284139202, 2.27238370722428728570015712288, 3.54574519606089903353001249150, 3.71822253537069739639390686069, 4.32773350578479900565119001608, 4.64849576847169223614699526425, 5.30537393915954566746162190631, 6.14620005069006563789988203374, 6.16116790751690085616531757265, 6.91673575142178820615272805521, 7.27965583167809929942184453108, 7.64537707540215722485320944275, 8.319627980737908137853866947983, 8.647026888430366294854058821975
