| L(s) = 1 | − 32·4-s − 72·9-s + 720·11-s + 640·16-s − 2.44e3·29-s + 2.30e3·36-s − 2.30e4·44-s + 1.53e3·49-s − 1.02e4·64-s − 4.30e4·71-s − 2.55e4·79-s − 2.24e4·81-s − 5.18e4·99-s + 2.69e4·109-s + 7.83e4·116-s + 2.15e5·121-s + 127-s + 131-s + 137-s + 139-s − 4.60e4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.22e5·169-s + ⋯ |
| L(s) = 1 | − 2·4-s − 8/9·9-s + 5.95·11-s + 5/2·16-s − 2.91·29-s + 16/9·36-s − 11.9·44-s + 0.638·49-s − 5/2·64-s − 8.54·71-s − 4.09·79-s − 3.41·81-s − 5.28·99-s + 2.27·109-s + 5.82·116-s + 14.7·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 2.22·144-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 7.80·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.01482186731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01482186731\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 1532 T^{2} - 58266 p^{2} T^{4} - 1532 p^{8} T^{6} + p^{16} T^{8} \) |
| good | 3 | \( ( 1 + 4 p^{2} T^{2} + 1462 p^{2} T^{4} + 4 p^{10} T^{6} + p^{16} T^{8} )^{2} \) |
| 11 | \( ( 1 - 180 T + 27014 T^{2} - 180 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 13 | \( ( 1 + 111460 T^{2} + 4736358630 T^{4} + 111460 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 + 217348 T^{2} + 25123879686 T^{4} + 217348 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 19 | \( ( 1 - 150436 T^{2} + 9183572838 T^{4} - 150436 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 23 | \( ( 1 - 893500 T^{2} + 344645803014 T^{4} - 893500 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 29 | \( ( 1 + 612 T + 1092326 T^{2} + 612 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1703428 T^{2} + 1577900136966 T^{4} - 1703428 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 - 5412668 T^{2} + 13995111207750 T^{4} - 5412668 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 41 | \( ( 1 - 7851268 T^{2} + 31379741059206 T^{4} - 7851268 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 43 | \( ( 1 - 9901436 T^{2} + 44360540905926 T^{4} - 9901436 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 47 | \( ( 1 + 10858756 T^{2} + 58650967963398 T^{4} + 10858756 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 53 | \( ( 1 - 10270012 T^{2} + 78439997736006 T^{4} - 10270012 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 59 | \( ( 1 - 42750244 T^{2} + 750538168343526 T^{4} - 42750244 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 61 | \( ( 1 - 28277476 T^{2} + 401277478909158 T^{4} - 28277476 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 - 79140092 T^{2} + 2377741700101446 T^{4} - 79140092 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 + 10764 T + 70144454 T^{2} + 10764 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 73 | \( ( 1 + 62983684 T^{2} + 1969418053172358 T^{4} + 62983684 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 + 6388 T + 73521798 T^{2} + 6388 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 83 | \( ( 1 + 129024292 T^{2} + 7865129580692070 T^{4} + 129024292 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 89 | \( ( 1 - 228854020 T^{2} + 20924407354852230 T^{4} - 228854020 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 + 150468100 T^{2} + 16842128301249030 T^{4} + 150468100 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.32807161142433165180827338568, −4.28296523788016589261941946221, −4.18406527258123410514580566963, −3.98389966101053609429323562737, −3.62106583496691192548617198816, −3.60513103015310419344860469538, −3.56571647228029618464564628872, −3.55597592250143701900998229229, −3.54561279357128957345943808537, −3.05417708760987828559336972163, −2.79144840505209130692477597692, −2.74062595703931528771417910955, −2.58859716412148781077706092180, −2.20636913562419107319724787171, −2.15185184483863171790628346166, −1.78288360297655346299198806180, −1.37206095782300106406076202168, −1.35954299808308929008308289622, −1.25951631873138198500016754920, −1.23983567507322747794766459944, −1.17189348856949464485209401782, −1.14878450057713696975758715938, −0.28477967453564112481191636218, −0.13270559716722082827945837586, −0.03512007377958707304507088347,
0.03512007377958707304507088347, 0.13270559716722082827945837586, 0.28477967453564112481191636218, 1.14878450057713696975758715938, 1.17189348856949464485209401782, 1.23983567507322747794766459944, 1.25951631873138198500016754920, 1.35954299808308929008308289622, 1.37206095782300106406076202168, 1.78288360297655346299198806180, 2.15185184483863171790628346166, 2.20636913562419107319724787171, 2.58859716412148781077706092180, 2.74062595703931528771417910955, 2.79144840505209130692477597692, 3.05417708760987828559336972163, 3.54561279357128957345943808537, 3.55597592250143701900998229229, 3.56571647228029618464564628872, 3.60513103015310419344860469538, 3.62106583496691192548617198816, 3.98389966101053609429323562737, 4.18406527258123410514580566963, 4.28296523788016589261941946221, 4.32807161142433165180827338568
Plot not available for L-functions of degree greater than 10.
