| L(s) = 1 | − 1.67e3·16-s + 3.26e4·31-s + 1.39e5·61-s − 5.90e4·81-s + 6.44e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 1.63·16-s + 6.09·31-s + 4.79·61-s − 81-s + 4·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s + 1.28e−6·227-s + 1.26e−6·229-s + 1.20e−6·233-s + 1.13e−6·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.670901197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.670901197\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{10} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 1673 T^{4} + p^{20} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 1820608576898 T^{4} + p^{20} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 269302 T^{2} + p^{10} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37064501443298 T^{4} + p^{20} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8152 T + p^{5} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 7974026877035902 T^{4} + p^{20} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 349504859353650098 T^{4} + p^{20} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 34802 T + p^{5} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 1245148702 T^{2} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 24050685636765349102 T^{4} + p^{20} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + p^{10} T^{4} )^{2} \) |
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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
−9.901862648173814374069129823218, −9.698878712724274750149955983925, −9.131385818145902954813262129270, −8.645818789051313835842614147318, −8.400395557732094386339150552223, −8.359870140284678993177183322494, −8.210747697144371775479682168056, −7.41771739610932976842406398687, −7.16606077458198679580541132621, −7.01467126915851242576323622584, −6.39163235979324175720705534518, −6.21425176569186591866997277062, −6.20488896473297388638427146390, −5.44657938468985319099827460879, −4.85377012083625279332668605485, −4.75094301488601034777153184951, −4.56123800504510355440798294212, −3.81084782390231776237438994288, −3.70304942843874074671155723165, −2.72448235849727979760634240531, −2.52700735184757943846513134356, −2.39650178642167174664423611789, −1.36530432482200037396190724604, −0.877850360441859825708411064560, −0.46942097395760816159121276717,
0.46942097395760816159121276717, 0.877850360441859825708411064560, 1.36530432482200037396190724604, 2.39650178642167174664423611789, 2.52700735184757943846513134356, 2.72448235849727979760634240531, 3.70304942843874074671155723165, 3.81084782390231776237438994288, 4.56123800504510355440798294212, 4.75094301488601034777153184951, 4.85377012083625279332668605485, 5.44657938468985319099827460879, 6.20488896473297388638427146390, 6.21425176569186591866997277062, 6.39163235979324175720705534518, 7.01467126915851242576323622584, 7.16606077458198679580541132621, 7.41771739610932976842406398687, 8.210747697144371775479682168056, 8.359870140284678993177183322494, 8.400395557732094386339150552223, 8.645818789051313835842614147318, 9.131385818145902954813262129270, 9.698878712724274750149955983925, 9.901862648173814374069129823218
