| L(s) = 1 | − 2·2-s + 4-s − 2·5-s + 4·7-s + 2·8-s − 9-s + 4·10-s − 8·11-s − 4·13-s − 8·14-s − 4·16-s + 2·18-s + 12·19-s − 2·20-s + 16·22-s − 2·23-s + 4·25-s + 8·26-s + 4·28-s + 2·29-s − 12·31-s + 2·32-s − 8·35-s − 36-s + 12·37-s − 24·38-s − 4·40-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s + 0.707·8-s − 1/3·9-s + 1.26·10-s − 2.41·11-s − 1.10·13-s − 2.13·14-s − 16-s + 0.471·18-s + 2.75·19-s − 0.447·20-s + 3.41·22-s − 0.417·23-s + 4/5·25-s + 1.56·26-s + 0.755·28-s + 0.371·29-s − 2.15·31-s + 0.353·32-s − 1.35·35-s − 1/6·36-s + 1.97·37-s − 3.89·38-s − 0.632·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1612368285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1612368285\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 2 T - 12 T^{3} - 29 T^{4} - 12 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 19 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 12 p T^{5} - 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2 T - 48 T^{2} + 12 T^{3} + 1747 T^{4} + 12 p T^{5} - 48 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 10 T + 21 T^{2} - 30 T^{3} + 460 T^{4} - 30 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 50 T^{2} + 96 T^{3} + 795 T^{4} + 96 p T^{5} + 50 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 14 T + 60 T^{2} + 588 T^{3} + 7075 T^{4} + 588 p T^{5} + 60 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T + 18 T^{2} + 432 T^{3} - 3653 T^{4} + 432 p T^{5} + 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 55 T^{2} - 456 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14 T + 88 T^{2} + 196 T^{3} - 4013 T^{4} + 196 p T^{5} + 88 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T - 115 T^{2} + 12 T^{3} + 11600 T^{4} + 12 p T^{5} - 115 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 16 T + 78 T^{2} - 576 T^{3} + 8467 T^{4} - 576 p T^{5} + 78 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 14 T + 29 T^{2} + 294 T^{3} + 8252 T^{4} + 294 p T^{5} + 29 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 18 T + 77 T^{2} - 954 T^{3} + 20172 T^{4} - 954 p T^{5} + 77 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−12.32155412043741912722116531154, −11.68127102739973871787790019888, −11.45864257037850880312151225599, −11.34993850722000104369126884548, −11.32098951143918327540100152469, −10.44712860429574289879660496629, −10.38455434275363389768523277618, −10.04358212902967516378340566452, −9.758188665870075328531043182577, −9.217517298351656496368033783061, −9.168284283031185434774753366717, −8.262432187883424117866718095399, −8.189665066694201113063771895453, −7.999607704626304097391748492245, −7.81723990346239717137102540570, −7.30170867762992245211796335260, −7.15013860187141160130065522007, −6.49219872856213178823662708925, −5.37304044904848557543964142803, −5.36273208423730830468980267546, −4.89459253855276575057430288772, −4.76394696203279592635746947386, −3.58445348312736865712644994143, −3.06943908801571298717436507810, −2.03226124301764104537415046840,
2.03226124301764104537415046840, 3.06943908801571298717436507810, 3.58445348312736865712644994143, 4.76394696203279592635746947386, 4.89459253855276575057430288772, 5.36273208423730830468980267546, 5.37304044904848557543964142803, 6.49219872856213178823662708925, 7.15013860187141160130065522007, 7.30170867762992245211796335260, 7.81723990346239717137102540570, 7.999607704626304097391748492245, 8.189665066694201113063771895453, 8.262432187883424117866718095399, 9.168284283031185434774753366717, 9.217517298351656496368033783061, 9.758188665870075328531043182577, 10.04358212902967516378340566452, 10.38455434275363389768523277618, 10.44712860429574289879660496629, 11.32098951143918327540100152469, 11.34993850722000104369126884548, 11.45864257037850880312151225599, 11.68127102739973871787790019888, 12.32155412043741912722116531154
