| L(s) = 1 | − 4·5-s + 16·19-s + 8·25-s + 8·29-s + 16·31-s − 24·41-s − 2·49-s − 16·59-s − 24·61-s − 24·71-s − 8·79-s − 18·81-s + 40·89-s − 64·95-s − 24·101-s + 8·109-s + 4·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + 149-s + 151-s − 64·155-s + 157-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 3.67·19-s + 8/5·25-s + 1.48·29-s + 2.87·31-s − 3.74·41-s − 2/7·49-s − 2.08·59-s − 3.07·61-s − 2.84·71-s − 0.900·79-s − 2·81-s + 4.23·89-s − 6.56·95-s − 2.38·101-s + 0.766·109-s + 4/11·121-s − 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.093468987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093468987\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 498 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 998 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 5798 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 24198 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 8838 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.47496136319428939972397312722, −6.22563502152465313739232001146, −5.93486353247760404995772620574, −5.81708238131126741238473586737, −5.70698439708876773513935393030, −5.25959068825765931601914490896, −4.93954927161212599939754421930, −4.88704124637001824397221639204, −4.74406624879041187935813394233, −4.58685632320739537074872622753, −4.42298828529723871609278316179, −4.04525917113415770651268004492, −3.77600919673265572612566353278, −3.51195646023694999616386957369, −3.25967745697196948144728411301, −3.04585413755205914392059975522, −2.96073466728549370859629088109, −2.89944972508436259973046230382, −2.61930405467392647325283009475, −1.78669590156165920953865237500, −1.64564527352812898807293491960, −1.47458733684502290564505463431, −0.964004370579141966656370610921, −0.804201497247775473838380407049, −0.21258566062777738064442856662,
0.21258566062777738064442856662, 0.804201497247775473838380407049, 0.964004370579141966656370610921, 1.47458733684502290564505463431, 1.64564527352812898807293491960, 1.78669590156165920953865237500, 2.61930405467392647325283009475, 2.89944972508436259973046230382, 2.96073466728549370859629088109, 3.04585413755205914392059975522, 3.25967745697196948144728411301, 3.51195646023694999616386957369, 3.77600919673265572612566353278, 4.04525917113415770651268004492, 4.42298828529723871609278316179, 4.58685632320739537074872622753, 4.74406624879041187935813394233, 4.88704124637001824397221639204, 4.93954927161212599939754421930, 5.25959068825765931601914490896, 5.70698439708876773513935393030, 5.81708238131126741238473586737, 5.93486353247760404995772620574, 6.22563502152465313739232001146, 6.47496136319428939972397312722
