| L(s) = 1 | + 3-s − 7-s + 3·9-s − 3·11-s − 2·13-s − 3·17-s + 7·19-s − 21-s − 3·23-s + 8·27-s − 3·29-s − 8·31-s − 3·33-s − 7·37-s − 2·39-s + 9·41-s + 11·43-s + 7·49-s − 3·51-s + 12·53-s + 7·57-s + 3·59-s − 11·61-s − 3·63-s − 7·67-s − 3·69-s + 3·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 9-s − 0.904·11-s − 0.554·13-s − 0.727·17-s + 1.60·19-s − 0.218·21-s − 0.625·23-s + 1.53·27-s − 0.557·29-s − 1.43·31-s − 0.522·33-s − 1.15·37-s − 0.320·39-s + 1.40·41-s + 1.67·43-s + 49-s − 0.420·51-s + 1.64·53-s + 0.927·57-s + 0.390·59-s − 1.40·61-s − 0.377·63-s − 0.855·67-s − 0.361·69-s + 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.244036854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.244036854\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−9.785803643992299702396848442999, −9.304172431401926771816539964901, −9.033563996065757948606504478478, −8.967458748663186278299827354818, −7.965849494779697797228370837623, −7.907114027005011922715909802333, −7.38643085355063515894707010690, −7.12931504747046084822208419280, −6.77973280573441567497834336456, −6.07663515344907385981297737030, −5.55283733182973576689413372161, −5.34793190737737350645560904208, −4.67721241071233470988239253944, −4.27262866591144054311350543833, −3.69952278497384691554750495843, −3.32398952127472321311962415047, −2.48665563328583161830652169178, −2.39466551979178135506446612394, −1.50964351999751680811038581088, −0.61058087422623416702052954854,
0.61058087422623416702052954854, 1.50964351999751680811038581088, 2.39466551979178135506446612394, 2.48665563328583161830652169178, 3.32398952127472321311962415047, 3.69952278497384691554750495843, 4.27262866591144054311350543833, 4.67721241071233470988239253944, 5.34793190737737350645560904208, 5.55283733182973576689413372161, 6.07663515344907385981297737030, 6.77973280573441567497834336456, 7.12931504747046084822208419280, 7.38643085355063515894707010690, 7.907114027005011922715909802333, 7.965849494779697797228370837623, 8.967458748663186278299827354818, 9.033563996065757948606504478478, 9.304172431401926771816539964901, 9.785803643992299702396848442999
