| L(s) = 1 | − 2·2-s − 13·4-s + 28·8-s + 34·9-s + 140·13-s + 21·16-s + 112·17-s − 68·18-s − 112·19-s + 20·25-s − 280·26-s + 58·32-s − 224·34-s − 442·36-s + 224·38-s − 520·43-s + 952·47-s + 842·49-s − 40·50-s − 1.82e3·52-s − 576·53-s − 1.17e3·59-s + 759·64-s − 240·67-s − 1.45e3·68-s + 952·72-s + 1.45e3·76-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.62·4-s + 1.23·8-s + 1.25·9-s + 2.98·13-s + 0.328·16-s + 1.59·17-s − 0.890·18-s − 1.35·19-s + 4/25·25-s − 2.11·26-s + 0.320·32-s − 1.12·34-s − 2.04·36-s + 0.956·38-s − 1.84·43-s + 2.95·47-s + 2.45·49-s − 0.113·50-s − 4.85·52-s − 1.49·53-s − 2.59·59-s + 1.48·64-s − 0.437·67-s − 2.59·68-s + 1.55·72-s + 2.19·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83521 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83521 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5787377548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5787377548\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $D_{4}$ | \( 1 - 112 T + 382 p T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 2 | $D_{4}$ | \( ( 1 + T + p^{3} T^{2} + p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 3 | $C_2^2 \wr C_2$ | \( 1 - 34 T^{2} + 1450 T^{4} - 34 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 4 p T^{2} + 12342 T^{4} - 4 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 842 T^{2} + 410922 T^{4} - 842 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 1698 T^{2} + 3550826 T^{4} - 1698 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 70 T + 4002 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 20346 T^{2} + 209323274 T^{4} - 20346 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 74036 T^{2} + 2514340758 T^{4} - 74036 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 114890 T^{2} + 5074759530 T^{4} - 114890 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 116372 T^{2} + 7903483062 T^{4} - 116372 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 158724 T^{2} + 15178105958 T^{4} - 158724 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 260 T + 128262 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 476 T + 257822 T^{2} - 476 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 288 T + 248662 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 588 T + 335494 T^{2} + 588 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 425396 T^{2} + 97219598358 T^{4} - 425396 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 120 T + 571334 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 1379018 T^{2} + 731023514346 T^{4} - 1379018 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 55196 T^{2} + 301853502630 T^{4} + 55196 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 1282922 T^{2} + 849811865706 T^{4} - 1282922 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 1428 T + 1595158 T^{2} + 1428 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 994 T + 1642394 T^{2} - 994 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 3375140 T^{2} + 4511732954310 T^{4} - 3375140 p^{6} T^{6} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.79662102442458929136121135343, −13.64459895664640960911155622437, −13.42761283186875752180698191079, −12.80989534907016739716398128188, −12.69245070233277704580363555412, −12.28563232661045129803580875266, −11.75212381344530589661965658315, −11.10281217338752786649014441118, −10.94014436599962510429839631532, −10.30434893899211389086971043941, −10.19791111715273864048033247342, −9.709340471103488431391009543474, −9.164635252183473952165056881961, −8.716488189713775148320336662018, −8.699953750691681295823385858616, −8.325899958509834730509700289978, −7.56594401686642402426066623063, −7.24207365458814842960906234474, −6.22065577514056024710167293684, −6.16330868745957486881641457251, −5.31977091221502005946431867360, −4.37923220276666023327149617093, −4.20523770250516741168475357012, −3.46918795984664353324196041812, −1.22109800368859220164441475038,
1.22109800368859220164441475038, 3.46918795984664353324196041812, 4.20523770250516741168475357012, 4.37923220276666023327149617093, 5.31977091221502005946431867360, 6.16330868745957486881641457251, 6.22065577514056024710167293684, 7.24207365458814842960906234474, 7.56594401686642402426066623063, 8.325899958509834730509700289978, 8.699953750691681295823385858616, 8.716488189713775148320336662018, 9.164635252183473952165056881961, 9.709340471103488431391009543474, 10.19791111715273864048033247342, 10.30434893899211389086971043941, 10.94014436599962510429839631532, 11.10281217338752786649014441118, 11.75212381344530589661965658315, 12.28563232661045129803580875266, 12.69245070233277704580363555412, 12.80989534907016739716398128188, 13.42761283186875752180698191079, 13.64459895664640960911155622437, 13.79662102442458929136121135343
