| L(s) = 1 | + 4·3-s + 12·7-s + 8·9-s + 8·13-s − 16-s + 48·21-s − 8·25-s + 12·27-s + 8·31-s + 24·37-s + 32·39-s − 4·48-s + 72·49-s − 40·61-s + 96·63-s − 40·67-s − 4·73-s − 32·75-s + 23·81-s + 96·91-s + 32·93-s − 12·97-s + 20·103-s + 96·111-s − 12·112-s + 64·117-s + 8·121-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 4.53·7-s + 8/3·9-s + 2.21·13-s − 1/4·16-s + 10.4·21-s − 8/5·25-s + 2.30·27-s + 1.43·31-s + 3.94·37-s + 5.12·39-s − 0.577·48-s + 72/7·49-s − 5.12·61-s + 12.0·63-s − 4.88·67-s − 0.468·73-s − 3.69·75-s + 23/9·81-s + 10.0·91-s + 3.31·93-s − 1.21·97-s + 1.97·103-s + 9.11·111-s − 1.13·112-s + 5.91·117-s + 8/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.09775463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.09775463\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 1918 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.78014710802500698015668123632, −7.41367817642557327183195350854, −7.36326794087583361470572873689, −7.33162434879645579830435576238, −6.49046827055773506818304654802, −6.14575115115453145280815326271, −6.10378474590593660329266067126, −5.94709448879461028593545111621, −5.87087569914698392670761710036, −5.05998737123769107944921199391, −4.92102442814472909164924446242, −4.85146624321649127553217899578, −4.45413902013504883758164319173, −4.30012948125587578638801145364, −4.26858302669838491804884666841, −3.74138766187394430470124594675, −3.66184650476906617790581911581, −3.02260025143626698054979396847, −2.77582060923752359984034529320, −2.59066299140401673991094262870, −2.29778624401432072462176375136, −1.62631062377066928415029251134, −1.58375035993343888500925056961, −1.39486749739789990693725664237, −1.12989471900069496180830158803,
1.12989471900069496180830158803, 1.39486749739789990693725664237, 1.58375035993343888500925056961, 1.62631062377066928415029251134, 2.29778624401432072462176375136, 2.59066299140401673991094262870, 2.77582060923752359984034529320, 3.02260025143626698054979396847, 3.66184650476906617790581911581, 3.74138766187394430470124594675, 4.26858302669838491804884666841, 4.30012948125587578638801145364, 4.45413902013504883758164319173, 4.85146624321649127553217899578, 4.92102442814472909164924446242, 5.05998737123769107944921199391, 5.87087569914698392670761710036, 5.94709448879461028593545111621, 6.10378474590593660329266067126, 6.14575115115453145280815326271, 6.49046827055773506818304654802, 7.33162434879645579830435576238, 7.36326794087583361470572873689, 7.41367817642557327183195350854, 7.78014710802500698015668123632
