| L(s) = 1 | + 2·4-s − 8·7-s + 16-s + 8·25-s − 16·28-s − 12·31-s − 4·37-s + 56·43-s + 12·49-s + 12·61-s − 2·64-s − 20·67-s − 72·73-s − 20·79-s + 16·100-s + 60·103-s + 20·109-s − 8·112-s − 8·121-s − 24·124-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4-s − 3.02·7-s + 1/4·16-s + 8/5·25-s − 3.02·28-s − 2.15·31-s − 0.657·37-s + 8.53·43-s + 12/7·49-s + 1.53·61-s − 1/4·64-s − 2.44·67-s − 8.42·73-s − 2.25·79-s + 8/5·100-s + 5.91·103-s + 1.91·109-s − 0.755·112-s − 0.727·121-s − 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04308282595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04308282595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} - 105 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 34 T^{2} + 555 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 28 T^{2} + 495 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 14 T^{2} - 165 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 8 T^{2} - 894 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 6 T + 23 T^{2} + 66 T^{3} - 468 T^{4} + 66 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T - 53 T^{2} - 34 T^{3} + 1732 T^{4} - 34 p T^{5} - 53 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 7 T + p T^{2} )^{8} \) |
| 47 | \( 1 + 40 T^{2} - 626 T^{4} - 87680 T^{6} - 3459005 T^{8} - 87680 p^{2} T^{10} - 626 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 4 T^{2} + 4762 T^{4} + 41456 T^{6} + 14589091 T^{8} + 41456 p^{2} T^{10} + 4762 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 112 T^{2} + 9063 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 6 T + 131 T^{2} - 714 T^{3} + 11172 T^{4} - 714 p T^{5} + 131 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 10 T + 13 T^{2} - 470 T^{3} - 3620 T^{4} - 470 p T^{5} + 13 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 79107 T^{4} + 8280 p T^{5} + 662 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 70 p T^{5} - 65 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 68 T^{2} + 4566 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 32 T^{2} + 8254 T^{4} + 738304 T^{6} - 18580445 T^{8} + 738304 p^{2} T^{10} + 8254 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 334 T^{2} + 46419 T^{4} - 334 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.04028860507216640092781304137, −4.79968519774219734071919844863, −4.65677218590670936953521271234, −4.61163258924878344494799853041, −4.39119045265655951491120401214, −4.33089915590753635755573465160, −4.18672145210840021439404454344, −4.02184673082405158798424180172, −3.87550330384995738353840016764, −3.49926994953112822885726444830, −3.40981879657180536897900609000, −3.33064832592204535138309223376, −3.22905340678241951066975829929, −3.04042345292778083799062935256, −2.81855434303915579719796488249, −2.70371834603069729774990166611, −2.62373335655672969775321743652, −2.32297116541641727875467440390, −2.17357927810151831613887630906, −2.04013866467793594819661442217, −1.56594491512459227818753684601, −1.37216278210050611772860377394, −1.06959992285701514232957566219, −0.824009368639995085470008281421, −0.04669036181114440510663954777,
0.04669036181114440510663954777, 0.824009368639995085470008281421, 1.06959992285701514232957566219, 1.37216278210050611772860377394, 1.56594491512459227818753684601, 2.04013866467793594819661442217, 2.17357927810151831613887630906, 2.32297116541641727875467440390, 2.62373335655672969775321743652, 2.70371834603069729774990166611, 2.81855434303915579719796488249, 3.04042345292778083799062935256, 3.22905340678241951066975829929, 3.33064832592204535138309223376, 3.40981879657180536897900609000, 3.49926994953112822885726444830, 3.87550330384995738353840016764, 4.02184673082405158798424180172, 4.18672145210840021439404454344, 4.33089915590753635755573465160, 4.39119045265655951491120401214, 4.61163258924878344494799853041, 4.65677218590670936953521271234, 4.79968519774219734071919844863, 5.04028860507216640092781304137
Plot not available for L-functions of degree greater than 10.
