| L(s) = 1 | + 16·4-s − 76·7-s − 360·11-s + 192·16-s + 792·23-s + 100·25-s − 1.21e3·28-s − 1.22e3·29-s − 3.89e3·37-s + 3.68e3·43-s − 5.76e3·44-s + 2.12e3·49-s − 5.83e3·53-s + 2.04e3·64-s − 1.04e3·67-s + 2.15e4·71-s + 2.73e4·77-s + 1.27e4·79-s + 1.26e4·92-s + 1.60e3·100-s + 1.00e4·107-s − 1.34e4·109-s − 1.45e4·112-s + 7.99e3·113-s − 1.95e4·116-s + 4.31e4·121-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s − 1.55·7-s − 2.97·11-s + 3/4·16-s + 1.49·23-s + 4/25·25-s − 1.55·28-s − 1.45·29-s − 2.84·37-s + 1.99·43-s − 2.97·44-s + 0.883·49-s − 2.07·53-s + 1/2·64-s − 0.233·67-s + 4.27·71-s + 4.61·77-s + 2.04·79-s + 1.49·92-s + 4/25·100-s + 0.874·107-s − 1.13·109-s − 1.16·112-s + 0.625·113-s − 1.45·116-s + 2.94·121-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4081579509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4081579509\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $D_{4}$ | \( 1 + 76 T + 522 p T^{2} + 76 p^{4} T^{3} + p^{8} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 4 p^{2} T^{2} + 345702 T^{4} - 4 p^{10} T^{6} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 180 T + 27014 T^{2} + 180 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 111460 T^{2} + 4736358630 T^{4} - 111460 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 217348 T^{2} + 25123879686 T^{4} - 217348 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 150436 T^{2} + 9183572838 T^{4} - 150436 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 396 T + 525158 T^{2} - 396 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 612 T + 1092326 T^{2} + 612 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 1703428 T^{2} + 1577900136966 T^{4} - 1703428 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 1948 T + 4603686 T^{2} + 1948 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 7851268 T^{2} + 31379741059206 T^{4} - 7851268 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 1844 T + 6650886 T^{2} - 1844 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 10858756 T^{2} + 58650967963398 T^{4} - 10858756 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 2916 T + 9386534 T^{2} + 2916 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 42750244 T^{2} + 750538168343526 T^{4} - 42750244 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 28277476 T^{2} + 401277478909158 T^{4} - 28277476 p^{8} T^{6} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 524 T + 39707334 T^{2} + 524 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 10764 T + 70144454 T^{2} - 10764 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 62983684 T^{2} + 1969418053172358 T^{4} - 62983684 p^{8} T^{6} + p^{16} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 6388 T + 73521798 T^{2} - 6388 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 129024292 T^{2} + 7865129580692070 T^{4} - 129024292 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 228854020 T^{2} + 20924407354852230 T^{4} - 228854020 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 150468100 T^{2} + 16842128301249030 T^{4} - 150468100 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−9.217437212970804514526916057350, −8.962090284328010864256868710122, −8.494030533942232206174133083565, −8.034160282236519039159698580654, −7.88769530255336204924795225187, −7.71615814103085576785727121831, −7.42003878424317137510137912176, −6.88926381853978103221022852727, −6.82218418804549569691936674443, −6.63538474686786748383738725597, −6.05843102185493511964232569078, −5.90784860915019853898793226708, −5.27497072879183001086609257125, −5.17603086503947877366219934633, −5.17461616707136437463786101549, −4.54557904363092472722977676079, −3.77346145895336409002187617048, −3.44911198538655525267282512301, −3.35189973866600603935208795479, −2.77449418190176261271997302782, −2.41278239847599662098602258674, −2.26401559368709357436593989315, −1.54205239674827537974731464872, −0.75599968739793929085518967072, −0.14472429287910170808987195697,
0.14472429287910170808987195697, 0.75599968739793929085518967072, 1.54205239674827537974731464872, 2.26401559368709357436593989315, 2.41278239847599662098602258674, 2.77449418190176261271997302782, 3.35189973866600603935208795479, 3.44911198538655525267282512301, 3.77346145895336409002187617048, 4.54557904363092472722977676079, 5.17461616707136437463786101549, 5.17603086503947877366219934633, 5.27497072879183001086609257125, 5.90784860915019853898793226708, 6.05843102185493511964232569078, 6.63538474686786748383738725597, 6.82218418804549569691936674443, 6.88926381853978103221022852727, 7.42003878424317137510137912176, 7.71615814103085576785727121831, 7.88769530255336204924795225187, 8.034160282236519039159698580654, 8.494030533942232206174133083565, 8.962090284328010864256868710122, 9.217437212970804514526916057350
