| L(s) = 1 | + 4·2-s − 2·3-s + 10·4-s − 8·6-s − 2·7-s + 20·8-s + 3·9-s − 20·12-s − 8·14-s + 35·16-s + 12·18-s + 4·21-s − 40·24-s − 6·27-s − 20·28-s + 56·32-s + 30·36-s − 2·41-s + 16·42-s − 2·43-s − 70·48-s + 3·49-s − 24·54-s − 40·56-s − 6·63-s + 84·64-s + 2·67-s + ⋯ |
| L(s) = 1 | + 4·2-s − 2·3-s + 10·4-s − 8·6-s − 2·7-s + 20·8-s + 3·9-s − 20·12-s − 8·14-s + 35·16-s + 12·18-s + 4·21-s − 40·24-s − 6·27-s − 20·28-s + 56·32-s + 30·36-s − 2·41-s + 16·42-s − 2·43-s − 70·48-s + 3·49-s − 24·54-s − 40·56-s − 6·63-s + 84·64-s + 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(12.65159545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.65159545\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.41309620907686691632061984662, −5.98849585522427375038919657611, −5.85379846085590710231635614065, −5.68847091455104255482246287730, −5.52886430571236832410393137298, −5.34045594637151400816262619322, −5.19370257786938874377602238148, −4.96441171921821926305060403306, −4.78094655509316772459830014847, −4.62520744386184460652465717094, −4.36133506507771311799667529868, −4.00874107000753962397629509879, −3.85247093224444412905809693895, −3.80949000323466071605269263545, −3.60409136017536410054609815505, −3.39232547889014650728683080049, −3.11392807792812071065267307891, −2.91152825212158226549359492547, −2.72979899478129858395223708089, −2.07075257243076135103761956433, −2.04214823574157853483346036175, −1.97577363131567864816377018114, −1.60398469939576053752727578108, −1.14953641591110374745511598523, −0.72956131242624455038558864009,
0.72956131242624455038558864009, 1.14953641591110374745511598523, 1.60398469939576053752727578108, 1.97577363131567864816377018114, 2.04214823574157853483346036175, 2.07075257243076135103761956433, 2.72979899478129858395223708089, 2.91152825212158226549359492547, 3.11392807792812071065267307891, 3.39232547889014650728683080049, 3.60409136017536410054609815505, 3.80949000323466071605269263545, 3.85247093224444412905809693895, 4.00874107000753962397629509879, 4.36133506507771311799667529868, 4.62520744386184460652465717094, 4.78094655509316772459830014847, 4.96441171921821926305060403306, 5.19370257786938874377602238148, 5.34045594637151400816262619322, 5.52886430571236832410393137298, 5.68847091455104255482246287730, 5.85379846085590710231635614065, 5.98849585522427375038919657611, 6.41309620907686691632061984662
