| L(s) = 1 | − 2·9-s + 20·17-s − 12·41-s − 12·49-s + 60·73-s − 15·81-s − 4·89-s − 40·97-s + 60·113-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 40·153-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 4.85·17-s − 1.87·41-s − 1.71·49-s + 7.02·73-s − 5/3·81-s − 0.423·89-s − 4.06·97-s + 5.64·113-s − 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 3.23·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.718435989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.718435989\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 9 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.65796013786199096763894985201, −5.28255496174374061458410899699, −5.21575255290964703119622074324, −5.19805721875933334058833977313, −5.13714912767965398111122377055, −4.76418950159562243308517310872, −4.65243363819386399413904217610, −4.13137085054261703956134527303, −4.09616156559995928335734833014, −3.82466789507699082833568887705, −3.61820031724829737908661133363, −3.42626339794745513562100399929, −3.42619973582313701232089350359, −3.08859854051818452993117504779, −2.86387469650395216228188505519, −2.83903352746008440598830797123, −2.56550108748421399421322768128, −2.14899338190640335463029706497, −1.75778439389160199255860821542, −1.72963272069328250295150942371, −1.49612872681251295947700160826, −1.21075325720060870587730909063, −0.792449170917085375064566359672, −0.55563200358841687574981577083, −0.47862600607906270561338145322,
0.47862600607906270561338145322, 0.55563200358841687574981577083, 0.792449170917085375064566359672, 1.21075325720060870587730909063, 1.49612872681251295947700160826, 1.72963272069328250295150942371, 1.75778439389160199255860821542, 2.14899338190640335463029706497, 2.56550108748421399421322768128, 2.83903352746008440598830797123, 2.86387469650395216228188505519, 3.08859854051818452993117504779, 3.42619973582313701232089350359, 3.42626339794745513562100399929, 3.61820031724829737908661133363, 3.82466789507699082833568887705, 4.09616156559995928335734833014, 4.13137085054261703956134527303, 4.65243363819386399413904217610, 4.76418950159562243308517310872, 5.13714912767965398111122377055, 5.19805721875933334058833977313, 5.21575255290964703119622074324, 5.28255496174374061458410899699, 5.65796013786199096763894985201
