| L(s) = 1 | + 2·7-s − 3·11-s + 2·13-s + 6·17-s − 2·19-s + 6·23-s + 5·25-s − 6·29-s − 4·31-s + 8·37-s + 9·41-s + 43-s + 6·47-s + 7·49-s + 24·53-s + 3·59-s + 8·61-s − 5·67-s − 24·71-s + 22·73-s − 6·77-s − 4·79-s + 12·83-s − 12·89-s + 4·91-s − 5·97-s + 14·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.904·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s + 25-s − 1.11·29-s − 0.718·31-s + 1.31·37-s + 1.40·41-s + 0.152·43-s + 0.875·47-s + 49-s + 3.29·53-s + 0.390·59-s + 1.02·61-s − 0.610·67-s − 2.84·71-s + 2.57·73-s − 0.683·77-s − 0.450·79-s + 1.31·83-s − 1.27·89-s + 0.419·91-s − 0.507·97-s + 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.190221062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.190221062\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.413952841324163097615930753077, −9.057524599899958380258212951267, −8.575093211109862766728822233167, −8.571954221164352986571117239136, −7.79042020656272212454085421325, −7.57755437851135847301686674868, −7.23350114385932295842075927388, −6.94946662572253571492045894966, −6.15541313084689976978420961095, −5.73707815255641406110948342675, −5.48933806763423039282282233877, −5.19762583583069829640978499542, −4.40765636839452239207921215767, −4.28871165727223884298952398795, −3.51825212912594374608338265663, −3.19951819318163394143757783228, −2.33240302370886618173630476921, −2.27513085812152785197533481898, −1.09102071250634604603810883828, −0.841618103715855155454521688878,
0.841618103715855155454521688878, 1.09102071250634604603810883828, 2.27513085812152785197533481898, 2.33240302370886618173630476921, 3.19951819318163394143757783228, 3.51825212912594374608338265663, 4.28871165727223884298952398795, 4.40765636839452239207921215767, 5.19762583583069829640978499542, 5.48933806763423039282282233877, 5.73707815255641406110948342675, 6.15541313084689976978420961095, 6.94946662572253571492045894966, 7.23350114385932295842075927388, 7.57755437851135847301686674868, 7.79042020656272212454085421325, 8.571954221164352986571117239136, 8.575093211109862766728822233167, 9.057524599899958380258212951267, 9.413952841324163097615930753077
