| L(s) = 1 | − 6·3-s − 2·4-s + 8·7-s + 21·9-s + 12·12-s + 3·16-s + 4·19-s − 48·21-s + 25-s − 54·27-s − 16·28-s + 8·31-s − 42·36-s − 18·37-s − 30·43-s − 18·48-s + 30·49-s − 24·57-s + 168·63-s − 4·64-s − 14·67-s + 48·73-s − 6·75-s − 8·76-s − 6·79-s + 108·81-s + 96·84-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 4-s + 3.02·7-s + 7·9-s + 3.46·12-s + 3/4·16-s + 0.917·19-s − 10.4·21-s + 1/5·25-s − 10.3·27-s − 3.02·28-s + 1.43·31-s − 7·36-s − 2.95·37-s − 4.57·43-s − 2.59·48-s + 30/7·49-s − 3.17·57-s + 21.1·63-s − 1/2·64-s − 1.71·67-s + 5.61·73-s − 0.692·75-s − 0.917·76-s − 0.675·79-s + 12·81-s + 10.4·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7608299013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7608299013\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 5 T^{2} - 96 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 7 T^{2} - 240 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 46 T^{2} + 435 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 15 T + 118 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 26 T^{2} - 2805 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 106 T^{2} + 6195 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 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
−7.11623048525930037095570498431, −6.79214318529036132460915014727, −6.65982992034449909591070772219, −6.63595746566033341382607318293, −6.41919867791955489009609549501, −5.82105856294525961704063579047, −5.64105892572236102608251823618, −5.49658953498859178661231458970, −5.38912911507031353540836683683, −5.07404363898823721767224408476, −4.92889131350663383143320303175, −4.82059737240067259022960998464, −4.69179122198578913378321690766, −4.32829470791041273020123434879, −4.28828910219472451588349703818, −3.71367310408674505092759369359, −3.43579055931173407819118864271, −3.31704772930903909036161592609, −2.70946714017810608318406937086, −1.81047491374115316195886002699, −1.80936429355201335079613872650, −1.72096346352899877178739416606, −1.19487658969905343734895072287, −0.78757118907454973014975984036, −0.38965102614282853727009427972,
0.38965102614282853727009427972, 0.78757118907454973014975984036, 1.19487658969905343734895072287, 1.72096346352899877178739416606, 1.80936429355201335079613872650, 1.81047491374115316195886002699, 2.70946714017810608318406937086, 3.31704772930903909036161592609, 3.43579055931173407819118864271, 3.71367310408674505092759369359, 4.28828910219472451588349703818, 4.32829470791041273020123434879, 4.69179122198578913378321690766, 4.82059737240067259022960998464, 4.92889131350663383143320303175, 5.07404363898823721767224408476, 5.38912911507031353540836683683, 5.49658953498859178661231458970, 5.64105892572236102608251823618, 5.82105856294525961704063579047, 6.41919867791955489009609549501, 6.63595746566033341382607318293, 6.65982992034449909591070772219, 6.79214318529036132460915014727, 7.11623048525930037095570498431
