| L(s) = 1 | + 24·7-s − 24·11-s + 48·13-s − 24·17-s + 24·23-s + 20·31-s + 48·37-s + 192·41-s − 72·43-s − 144·47-s + 288·49-s − 120·53-s + 44·61-s + 24·67-s + 96·71-s + 48·73-s − 576·77-s − 9·81-s − 48·83-s + 1.15e3·91-s − 192·97-s + 240·101-s + 48·103-s + 360·107-s + 288·113-s − 576·119-s + 308·121-s + ⋯ |
| L(s) = 1 | + 24/7·7-s − 2.18·11-s + 3.69·13-s − 1.41·17-s + 1.04·23-s + 0.645·31-s + 1.29·37-s + 4.68·41-s − 1.67·43-s − 3.06·47-s + 5.87·49-s − 2.26·53-s + 0.721·61-s + 0.358·67-s + 1.35·71-s + 0.657·73-s − 7.48·77-s − 1/9·81-s − 0.578·83-s + 12.6·91-s − 1.97·97-s + 2.37·101-s + 0.466·103-s + 3.36·107-s + 2.54·113-s − 4.84·119-s + 2.54·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(13.10033413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.10033413\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2832 T^{3} + 23087 T^{4} - 2832 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 12 T + 62 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 48 T + 1152 T^{2} - 20640 T^{3} + 301679 T^{4} - 20640 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 6072 T^{3} + 126722 T^{4} + 6072 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 290 T^{2} - 30237 T^{4} - 290 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} + 8904 T^{3} - 534718 T^{4} + 8904 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2860 T^{2} + 3428358 T^{4} - 2860 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 10 T + 3 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 48 T + 1152 T^{2} - 21936 T^{3} - 414046 T^{4} - 21936 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 96 T + 5450 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 72 T + 2592 T^{2} + 174384 T^{3} + 11403839 T^{4} + 174384 p^{2} T^{5} + 2592 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 144 T + 10368 T^{2} + 551376 T^{3} + 26698082 T^{4} + 551376 p^{2} T^{5} + 10368 p^{4} T^{6} + 144 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 120 T + 7200 T^{2} + 345720 T^{3} + 16595138 T^{4} + 345720 p^{2} T^{5} + 7200 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 6062 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 22 T + 4107 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 103632 T^{3} + 37261007 T^{4} - 103632 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 48 T + 10442 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 48 T + 1152 T^{2} - 156720 T^{3} + 17060354 T^{4} - 156720 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 23228 T^{2} + 212771334 T^{4} - 23228 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 90480 T^{3} - 17933566 T^{4} + 90480 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 29668 T^{2} + 345034374 T^{4} - 29668 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 192 T + 18432 T^{2} + 2676864 T^{3} + 368210639 T^{4} + 2676864 p^{2} T^{5} + 18432 p^{4} T^{6} + 192 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.95115599319056031699720522720, −6.47005992699827904854359135159, −6.17083204078803232568139894073, −6.00741635413371555105430357854, −5.98526892603589423436870097890, −5.62954624808088068899789018115, −5.31397244772285644521515360168, −5.30302590746109365495226380176, −4.77894314175557407919985425430, −4.65223401450562193236547567918, −4.50014318182790598787890041256, −4.43405315298708943616371734190, −4.32901532342961410724303359198, −3.64393164463937301693299738157, −3.38172195274490193240427064525, −3.30760494315131761246799542378, −3.03964902260561882728975641963, −2.45254583355364213559014455058, −2.16099170220949301263563291237, −2.15807467152659401305002430488, −1.74226115028170509552854902819, −1.32707819818830284044071281363, −1.17585848464156141886523962950, −0.77802957580670613493260221548, −0.49468915295161580528199684779,
0.49468915295161580528199684779, 0.77802957580670613493260221548, 1.17585848464156141886523962950, 1.32707819818830284044071281363, 1.74226115028170509552854902819, 2.15807467152659401305002430488, 2.16099170220949301263563291237, 2.45254583355364213559014455058, 3.03964902260561882728975641963, 3.30760494315131761246799542378, 3.38172195274490193240427064525, 3.64393164463937301693299738157, 4.32901532342961410724303359198, 4.43405315298708943616371734190, 4.50014318182790598787890041256, 4.65223401450562193236547567918, 4.77894314175557407919985425430, 5.30302590746109365495226380176, 5.31397244772285644521515360168, 5.62954624808088068899789018115, 5.98526892603589423436870097890, 6.00741635413371555105430357854, 6.17083204078803232568139894073, 6.47005992699827904854359135159, 6.95115599319056031699720522720
