| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 9-s − 10-s + 14-s + 18-s − 19-s + 20-s + 25-s − 28-s − 31-s + 32-s − 35-s − 36-s + 38-s − 41-s − 45-s − 2·47-s + 49-s − 50-s + 59-s + 62-s + 63-s − 64-s + 8·67-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 9-s − 10-s + 14-s + 18-s − 19-s + 20-s + 25-s − 28-s − 31-s + 32-s − 35-s − 36-s + 38-s − 41-s − 45-s − 2·47-s + 49-s − 50-s + 59-s + 62-s + 63-s − 64-s + 8·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.206875636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206875636\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| 31 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) |
| 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4\times C_2$ | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4\times C_2$ | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_1$ | \( ( 1 - T )^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
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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
−6.29562406628143974526387470672, −6.15488745041482984981974772522, −5.99318525779222270885451073034, −5.46026693583999783089678021149, −5.43470052850309314295612016107, −5.35529795677303989672817467700, −5.01842824071235454354252655776, −4.99406453692623404059795355226, −4.79997606311949822150705248377, −4.32142152939224739289463711484, −4.17754510240933359065461175748, −3.73678868391412645445383440287, −3.67624682483074220333461290045, −3.56215987067673581996187921711, −3.42635082431809520570689859744, −2.90241564963806221096291873120, −2.72216739588345125031384659056, −2.48652585869984212165657783628, −2.44062459612844839770683031429, −1.94838404784633688570682754973, −1.92599520115765515993983048090, −1.68208873407973455978056305945, −1.23037206237178066326921688807, −0.68169912607888718613339467906, −0.66637170541311539055544294686,
0.66637170541311539055544294686, 0.68169912607888718613339467906, 1.23037206237178066326921688807, 1.68208873407973455978056305945, 1.92599520115765515993983048090, 1.94838404784633688570682754973, 2.44062459612844839770683031429, 2.48652585869984212165657783628, 2.72216739588345125031384659056, 2.90241564963806221096291873120, 3.42635082431809520570689859744, 3.56215987067673581996187921711, 3.67624682483074220333461290045, 3.73678868391412645445383440287, 4.17754510240933359065461175748, 4.32142152939224739289463711484, 4.79997606311949822150705248377, 4.99406453692623404059795355226, 5.01842824071235454354252655776, 5.35529795677303989672817467700, 5.43470052850309314295612016107, 5.46026693583999783089678021149, 5.99318525779222270885451073034, 6.15488745041482984981974772522, 6.29562406628143974526387470672
