| L(s) = 1 | + 63·4-s − 81·9-s − 680·11-s + 2.94e3·16-s − 1.78e3·19-s + 1.64e4·29-s − 4.99e3·31-s − 5.10e3·36-s + 3.96e4·41-s − 4.28e4·44-s − 2.40e3·49-s + 8.47e4·59-s + 2.95e4·61-s + 1.21e5·64-s + 2.91e4·71-s − 1.12e5·76-s + 4.54e3·79-s + 6.56e3·81-s + 2.34e5·89-s + 5.50e4·99-s − 2.17e5·101-s + 9.21e4·109-s + 1.03e6·116-s + 2.46e4·121-s − 3.14e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.96·4-s − 1/3·9-s − 1.69·11-s + 2.87·16-s − 1.13·19-s + 3.63·29-s − 0.932·31-s − 0.656·36-s + 3.68·41-s − 3.33·44-s − 1/7·49-s + 3.17·59-s + 1.01·61-s + 3.69·64-s + 0.685·71-s − 2.23·76-s + 0.0819·79-s + 1/9·81-s + 3.13·89-s + 0.564·99-s − 2.12·101-s + 0.743·109-s + 7.16·116-s + 0.153·121-s − 1.83·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.651056253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.651056253\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 63 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 340 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 536470 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2202910 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 892 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2683822 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8242 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2496 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 42687110 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 19834 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 3062810 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 378980830 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 836368486 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 42396 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14758 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2697441238 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14568 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1996967698 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2272 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6451961590 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 117286 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 17074640510 T^{2} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.34855154126651163910116529711, −10.21877312442153472369271278421, −9.539930802938821547542978563384, −8.841570237401453904916397195269, −8.201232510173380377435456814628, −8.093359423733256485403281274681, −7.62301333734901828672932142925, −7.07192354727656082623212264657, −6.55510443540717000154773896784, −6.38965058811131032132936348056, −5.60513849393895086389154288704, −5.47555024839076003730230663124, −4.68372193317830467201781981675, −4.05388294609623167481252015840, −3.28352992719996725955775250561, −2.60346247328115620667975010029, −2.49337970870526630662315142631, −2.07527145332434108218637031657, −0.937123339118542102148558613541, −0.66932771822843891191417486954,
0.66932771822843891191417486954, 0.937123339118542102148558613541, 2.07527145332434108218637031657, 2.49337970870526630662315142631, 2.60346247328115620667975010029, 3.28352992719996725955775250561, 4.05388294609623167481252015840, 4.68372193317830467201781981675, 5.47555024839076003730230663124, 5.60513849393895086389154288704, 6.38965058811131032132936348056, 6.55510443540717000154773896784, 7.07192354727656082623212264657, 7.62301333734901828672932142925, 8.093359423733256485403281274681, 8.201232510173380377435456814628, 8.841570237401453904916397195269, 9.539930802938821547542978563384, 10.21877312442153472369271278421, 10.34855154126651163910116529711
