| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 3·9-s + 2·12-s + 16-s − 12·17-s + 3·18-s − 8·19-s + 2·24-s + 25-s + 4·27-s + 32-s − 12·34-s + 3·36-s − 8·38-s + 12·41-s + 16·43-s + 2·48-s + 49-s + 50-s − 24·51-s + 4·54-s − 16·57-s − 24·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 9-s + 0.577·12-s + 1/4·16-s − 2.91·17-s + 0.707·18-s − 1.83·19-s + 0.408·24-s + 1/5·25-s + 0.769·27-s + 0.176·32-s − 2.05·34-s + 1/2·36-s − 1.29·38-s + 1.87·41-s + 2.43·43-s + 0.288·48-s + 1/7·49-s + 0.141·50-s − 3.36·51-s + 0.544·54-s − 2.11·57-s − 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1411200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1411200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.627609845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.627609845\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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
−7.978515145485646069187735022963, −7.33108854902409788194436714013, −7.27965583167809929942184453108, −6.41768772317016312336259897203, −6.20158409364579029263712797634, −6.14620005069006563789988203374, −4.92743542349778859885880619567, −4.64849576847169223614699526425, −4.44325577330637420989494109531, −3.71822253537069739639390686069, −3.55247748731974953241856335915, −2.48439822600300826951031607624, −2.18567106423834991001318834218, −2.16105217917237072156284139202, −0.73883825417033453364744445337,
0.73883825417033453364744445337, 2.16105217917237072156284139202, 2.18567106423834991001318834218, 2.48439822600300826951031607624, 3.55247748731974953241856335915, 3.71822253537069739639390686069, 4.44325577330637420989494109531, 4.64849576847169223614699526425, 4.92743542349778859885880619567, 6.14620005069006563789988203374, 6.20158409364579029263712797634, 6.41768772317016312336259897203, 7.27965583167809929942184453108, 7.33108854902409788194436714013, 7.978515145485646069187735022963
