| L(s) = 1 | + 4·7-s − 2·9-s + 8·17-s − 4·23-s − 6·25-s − 8·31-s + 16·41-s + 10·49-s − 8·63-s + 32·71-s + 24·73-s + 2·81-s + 32·89-s + 16·97-s + 24·103-s + 36·113-s + 32·119-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·153-s + 157-s − 16·161-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 2/3·9-s + 1.94·17-s − 0.834·23-s − 6/5·25-s − 1.43·31-s + 2.49·41-s + 10/7·49-s − 1.00·63-s + 3.79·71-s + 2.80·73-s + 2/9·81-s + 3.39·89-s + 1.62·97-s + 2.36·103-s + 3.38·113-s + 2.93·119-s − 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.29·153-s + 0.0798·157-s − 1.26·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.912872294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.912872294\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T^{2} + 2 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 6 T^{2} + 42 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 24 T^{2} + 318 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 38 T^{2} + 682 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 66 T^{2} + 1794 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 76 T^{2} + 2854 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 108 T^{2} + 5382 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 152 T^{2} + 9406 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 132 T^{2} + 8886 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 178 T^{2} + 14050 T^{4} + 178 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 230 T^{2} + 20650 T^{4} + 230 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 112 T^{2} + 8782 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 210 T^{2} + 23970 T^{4} + 210 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.58762574954538226292727188939, −6.14801460805488571662274418373, −6.07806740249784940116968354743, −5.99583877030064121998982458951, −5.93896759712490311872045842441, −5.40962636356547148044290285215, −5.22811483155944353768458301191, −5.20156333904630845714755225082, −5.01155102445970823984750008837, −4.65894318348720601120656671715, −4.53888768328166144068059511762, −4.13931693507848809426484022139, −3.87719977522342135836036584618, −3.68422365984952772772831428934, −3.44851915514365978278468556975, −3.42777478293824263838405136884, −3.07573611236688404209844162154, −2.46730107205941771081489948298, −2.28340221704207430623361418785, −2.13586314404084476079001678233, −1.99062849331431991659766749569, −1.60024953263741066708007673725, −0.990837736067871507166422003521, −0.833098801920312158407662725750, −0.52399927878331299383355506105,
0.52399927878331299383355506105, 0.833098801920312158407662725750, 0.990837736067871507166422003521, 1.60024953263741066708007673725, 1.99062849331431991659766749569, 2.13586314404084476079001678233, 2.28340221704207430623361418785, 2.46730107205941771081489948298, 3.07573611236688404209844162154, 3.42777478293824263838405136884, 3.44851915514365978278468556975, 3.68422365984952772772831428934, 3.87719977522342135836036584618, 4.13931693507848809426484022139, 4.53888768328166144068059511762, 4.65894318348720601120656671715, 5.01155102445970823984750008837, 5.20156333904630845714755225082, 5.22811483155944353768458301191, 5.40962636356547148044290285215, 5.93896759712490311872045842441, 5.99583877030064121998982458951, 6.07806740249784940116968354743, 6.14801460805488571662274418373, 6.58762574954538226292727188939
