| L(s) = 1 | + 3-s − 2·4-s − 2·9-s − 6·11-s − 2·12-s + 10·13-s + 4·16-s − 12·23-s + 25-s − 5·27-s − 6·33-s + 4·36-s + 4·37-s + 10·39-s + 12·44-s + 18·47-s + 4·48-s + 49-s − 20·52-s + 16·61-s − 8·64-s − 12·69-s + 4·73-s + 75-s + 81-s + 24·83-s + 24·92-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 2/3·9-s − 1.80·11-s − 0.577·12-s + 2.77·13-s + 16-s − 2.50·23-s + 1/5·25-s − 0.962·27-s − 1.04·33-s + 2/3·36-s + 0.657·37-s + 1.60·39-s + 1.80·44-s + 2.62·47-s + 0.577·48-s + 1/7·49-s − 2.77·52-s + 2.04·61-s − 64-s − 1.44·69-s + 0.468·73-s + 0.115·75-s + 1/9·81-s + 2.63·83-s + 2.50·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.283668432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283668432\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.034294939496608256127581309995, −8.679498853252053444462430241617, −8.153192339337884543589566310226, −7.980947296320422939728903060714, −7.67755521010834815156907872378, −6.60444721122988741193728663303, −6.06485042125071096226940610377, −5.54008340383010136479817620846, −5.47907564040983295563221596641, −4.41617775358459899843679315087, −3.82466491546191947603335262315, −3.61689003045514298236484607491, −2.72164418406817349261491719715, −2.03544754569172366495048437388, −0.72587888295584277318850692735,
0.72587888295584277318850692735, 2.03544754569172366495048437388, 2.72164418406817349261491719715, 3.61689003045514298236484607491, 3.82466491546191947603335262315, 4.41617775358459899843679315087, 5.47907564040983295563221596641, 5.54008340383010136479817620846, 6.06485042125071096226940610377, 6.60444721122988741193728663303, 7.67755521010834815156907872378, 7.980947296320422939728903060714, 8.153192339337884543589566310226, 8.679498853252053444462430241617, 9.034294939496608256127581309995
