| L(s) = 1 | + 2·2-s + 12·4-s + 96·7-s + 104·8-s + 192·14-s − 592·16-s − 200·17-s − 2.33e3·23-s + 7.02e3·25-s + 1.15e3·28-s − 1.29e4·31-s − 1.63e3·32-s − 400·34-s + 4.56e3·41-s − 4.67e3·46-s + 5.47e4·47-s − 2.40e4·49-s + 1.40e4·50-s + 9.98e3·56-s − 2.58e4·62-s − 1.36e4·64-s − 2.40e3·68-s − 2.06e5·71-s + 3.99e4·73-s − 2.47e5·79-s + 9.13e3·82-s + 8.46e4·89-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 3/8·4-s + 0.740·7-s + 0.574·8-s + 0.261·14-s − 0.578·16-s − 0.167·17-s − 0.920·23-s + 2.24·25-s + 0.277·28-s − 2.41·31-s − 0.281·32-s − 0.0593·34-s + 0.424·41-s − 0.325·46-s + 3.61·47-s − 1.43·49-s + 0.795·50-s + 0.425·56-s − 0.854·62-s − 0.416·64-s − 0.0629·68-s − 4.86·71-s + 0.877·73-s − 4.46·79-s + 0.150·82-s + 1.13·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.046907014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.046907014\) |
| \(L(\frac{7}{2})\) |
|
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 | $D_{4}$ | \( 1 - p T - p^{3} T^{2} - p^{6} T^{3} + p^{10} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 7028 T^{2} + 24404246 T^{4} - 7028 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 48 T + 15502 T^{2} - 48 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 296436 T^{2} + 49128544726 T^{4} - 296436 p^{10} T^{6} + p^{20} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 894228 T^{2} + 396323515894 T^{4} - 894228 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 100 T + 2767462 T^{2} + 100 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 6794580 T^{2} + 21506967947254 T^{4} - 6794580 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 1168 T + 10055470 T^{2} + 1168 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 31255380 T^{2} + 976386653995702 T^{4} - 31255380 p^{10} T^{6} + p^{20} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 6464 T + 65013054 T^{2} + 6464 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 241262580 T^{2} + 24149916431784598 T^{4} - 241262580 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2284 T + 146603254 T^{2} - 2284 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 10845276 p T^{2} + 96250708269010006 T^{4} - 10845276 p^{11} T^{6} + p^{20} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 27360 T + 591338206 T^{2} - 27360 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 1039152180 T^{2} + 595616955270391126 T^{4} - 1039152180 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 1537424180 T^{2} + 1399694789142612374 T^{4} - 1537424180 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 741098540 T^{2} + 1478044222094100534 T^{4} + 741098540 p^{10} T^{6} + p^{20} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 1366835860 T^{2} + 3274116308996825526 T^{4} - 1366835860 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 103344 T + 6217736974 T^{2} + 103344 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 19988 T + 2541602870 T^{2} - 19988 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 123936 T + 9855929374 T^{2} + 123936 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 10047855188 T^{2} + 55381071937674414326 T^{4} - 10047855188 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 42316 T + 4292401174 T^{2} - 42316 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 49788 T + 16391371462 T^{2} + 49788 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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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
−10.02914172095322099303795085352, −9.465466649755616828664738393133, −9.043744216709335483438246310223, −8.851080588572328010978904175791, −8.782660581432358504806540471062, −8.379450175054593207069688218896, −7.84349232555600511343186477085, −7.38852820630246332979862146382, −7.36678144540023270324973655727, −7.13332485628211379854770230195, −6.70621577355986067336136797941, −6.20158317513742414029692872345, −5.73510017061929838734168179630, −5.68774764390022729672926368411, −5.16755724614367709776668500631, −4.63966170297773543294947510000, −4.43540967211811199900795638233, −4.12625700560265773822107495075, −3.60156404961193829120447934661, −2.95038301354742714323249942770, −2.65530714771546617907825849581, −2.06856297875103435786009563541, −1.53097344825621566784489922319, −1.19944625352295596993422618158, −0.25879601075407652369797870652,
0.25879601075407652369797870652, 1.19944625352295596993422618158, 1.53097344825621566784489922319, 2.06856297875103435786009563541, 2.65530714771546617907825849581, 2.95038301354742714323249942770, 3.60156404961193829120447934661, 4.12625700560265773822107495075, 4.43540967211811199900795638233, 4.63966170297773543294947510000, 5.16755724614367709776668500631, 5.68774764390022729672926368411, 5.73510017061929838734168179630, 6.20158317513742414029692872345, 6.70621577355986067336136797941, 7.13332485628211379854770230195, 7.36678144540023270324973655727, 7.38852820630246332979862146382, 7.84349232555600511343186477085, 8.379450175054593207069688218896, 8.782660581432358504806540471062, 8.851080588572328010978904175791, 9.043744216709335483438246310223, 9.465466649755616828664738393133, 10.02914172095322099303795085352
