| L(s) = 1 | − 2·3-s + 8·7-s + 9-s + 6·11-s − 16·21-s + 5·25-s − 4·27-s − 12·33-s − 16·37-s + 30·41-s + 12·47-s + 12·49-s − 12·53-s + 8·63-s − 2·67-s + 10·73-s − 10·75-s + 48·77-s − 4·81-s + 24·83-s + 6·99-s − 12·101-s + 6·107-s + 32·111-s − 11·121-s − 60·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3.02·7-s + 1/3·9-s + 1.80·11-s − 3.49·21-s + 25-s − 0.769·27-s − 2.08·33-s − 2.63·37-s + 4.68·41-s + 1.75·47-s + 12/7·49-s − 1.64·53-s + 1.00·63-s − 0.244·67-s + 1.17·73-s − 1.15·75-s + 5.47·77-s − 4/9·81-s + 2.63·83-s + 0.603·99-s − 1.19·101-s + 0.580·107-s + 3.03·111-s − 121-s − 5.41·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.676904185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.676904185\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| good | 3 | $D_{4}$ | \( ( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - p T^{2} + 9 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 11 | $D_{4}$ | \( ( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 633 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 162 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 30 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 77 T^{2} + 2493 T^{4} - 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 101 T^{2} + 4185 T^{4} - 101 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 1653 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 15 T + 133 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 13950 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 13 T^{2} + 5169 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + T + 87 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 5 T + 105 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 11973 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 20478 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.42218605794742695369514727858, −5.94018237902544048572204496104, −5.90268748923651741542234360450, −5.87913921238058549059820918427, −5.58485114066878824582000219282, −5.11516820878425889830739598101, −5.03204682960748304176726797506, −4.99654885446403163618001751027, −4.92005655641671058181415552312, −4.54322608771590392238942800965, −4.20588667612514579579003495724, −4.11310620312164614051509644756, −3.87434693572374988689400921228, −3.85261821329117389858783801633, −3.40074669212412105033369102525, −3.08725891661245331593820724504, −2.69546947526028342329291778800, −2.53663243174114953014976396468, −2.17971091804447593670885570866, −1.89280833782168542080038080207, −1.47573615914282718491903946366, −1.43851445032261960294945680959, −1.28648392105966870198274202782, −0.77704852772121285920396392283, −0.40491731873752843735159577121,
0.40491731873752843735159577121, 0.77704852772121285920396392283, 1.28648392105966870198274202782, 1.43851445032261960294945680959, 1.47573615914282718491903946366, 1.89280833782168542080038080207, 2.17971091804447593670885570866, 2.53663243174114953014976396468, 2.69546947526028342329291778800, 3.08725891661245331593820724504, 3.40074669212412105033369102525, 3.85261821329117389858783801633, 3.87434693572374988689400921228, 4.11310620312164614051509644756, 4.20588667612514579579003495724, 4.54322608771590392238942800965, 4.92005655641671058181415552312, 4.99654885446403163618001751027, 5.03204682960748304176726797506, 5.11516820878425889830739598101, 5.58485114066878824582000219282, 5.87913921238058549059820918427, 5.90268748923651741542234360450, 5.94018237902544048572204496104, 6.42218605794742695369514727858
