| L(s) = 1 | − 76·7-s + 36·9-s − 360·11-s − 792·23-s + 100·25-s − 1.22e3·29-s + 3.89e3·37-s − 3.68e3·43-s + 2.12e3·49-s − 5.83e3·53-s − 2.73e3·63-s + 1.04e3·67-s − 2.15e4·71-s + 2.73e4·77-s + 1.27e4·79-s − 1.18e4·81-s − 1.29e4·99-s + 1.00e4·107-s + 1.34e4·109-s − 7.99e3·113-s + 4.31e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.55·7-s + 4/9·9-s − 2.97·11-s − 1.49·23-s + 4/25·25-s − 1.45·29-s + 2.84·37-s − 1.99·43-s + 0.883·49-s − 2.07·53-s − 0.689·63-s + 0.233·67-s − 4.27·71-s + 4.61·77-s + 2.04·79-s − 1.80·81-s − 1.32·99-s + 0.874·107-s + 1.13·109-s − 0.625·113-s + 2.94·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.02310021134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02310021134\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 7 | $D_{4}$ | \( 1 + 76 T + 522 p T^{2} + 76 p^{4} T^{3} + p^{8} T^{4} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 4 p^{2} T^{2} + 1462 p^{2} T^{4} - 4 p^{10} T^{6} + p^{16} T^{8} \) |
| 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
−7.27842960720282969200106435676, −7.22956442189228326423656375129, −7.18491609757673151253641958585, −6.39171168200024475366341494446, −6.36879711092669481084023100718, −6.05322883155413170971135887510, −5.98006112728662921315653545448, −5.84060897829128582016217163864, −5.21897683441617812187078184472, −5.11526074909850544364981277598, −4.90993666607781902910890296665, −4.62639335634641481380409522840, −4.25480215881864952629281972348, −3.83152158347403036174882463062, −3.80063155314878208937102648217, −3.20615871474650411193239925419, −3.08404114633171324939053081703, −2.77350973455447052967581277789, −2.52611819754367760592639566194, −2.17283793279220840085404143029, −1.90507659047097910460333414335, −1.20198258363689457022897077023, −1.16083330769771848997412355039, −0.13716999809232537602215420346, −0.087406963958642979444619661139,
0.087406963958642979444619661139, 0.13716999809232537602215420346, 1.16083330769771848997412355039, 1.20198258363689457022897077023, 1.90507659047097910460333414335, 2.17283793279220840085404143029, 2.52611819754367760592639566194, 2.77350973455447052967581277789, 3.08404114633171324939053081703, 3.20615871474650411193239925419, 3.80063155314878208937102648217, 3.83152158347403036174882463062, 4.25480215881864952629281972348, 4.62639335634641481380409522840, 4.90993666607781902910890296665, 5.11526074909850544364981277598, 5.21897683441617812187078184472, 5.84060897829128582016217163864, 5.98006112728662921315653545448, 6.05322883155413170971135887510, 6.36879711092669481084023100718, 6.39171168200024475366341494446, 7.18491609757673151253641958585, 7.22956442189228326423656375129, 7.27842960720282969200106435676
