| L(s) = 1 | + 66·5-s − 176·7-s − 60·11-s − 658·13-s + 414·17-s − 956·19-s + 600·23-s + 1.23e3·25-s − 5.57e3·29-s + 3.59e3·31-s − 1.16e4·35-s − 8.45e3·37-s − 1.91e4·41-s − 1.33e4·43-s − 1.96e4·47-s + 1.41e4·49-s + 3.12e4·53-s − 3.96e3·55-s + 2.63e4·59-s − 3.10e4·61-s − 4.34e4·65-s + 1.68e4·67-s + 6.12e3·71-s − 2.55e4·73-s + 1.05e4·77-s − 7.44e4·79-s − 6.46e3·83-s + ⋯ |
| L(s) = 1 | + 1.18·5-s − 1.35·7-s − 0.149·11-s − 1.07·13-s + 0.347·17-s − 0.607·19-s + 0.236·23-s + 0.393·25-s − 1.23·29-s + 0.671·31-s − 1.60·35-s − 1.01·37-s − 1.78·41-s − 1.09·43-s − 1.29·47-s + 0.843·49-s + 1.52·53-s − 0.176·55-s + 0.985·59-s − 1.06·61-s − 1.27·65-s + 0.457·67-s + 0.144·71-s − 0.561·73-s + 0.202·77-s − 1.34·79-s − 0.103·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 66 T + p^{5} T^{2} \) |
| 7 | \( 1 + 176 T + p^{5} T^{2} \) |
| 11 | \( 1 + 60 T + p^{5} T^{2} \) |
| 13 | \( 1 + 658 T + p^{5} T^{2} \) |
| 17 | \( 1 - 414 T + p^{5} T^{2} \) |
| 19 | \( 1 + 956 T + p^{5} T^{2} \) |
| 23 | \( 1 - 600 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5574 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3592 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8458 T + p^{5} T^{2} \) |
| 41 | \( 1 + 19194 T + p^{5} T^{2} \) |
| 43 | \( 1 + 13316 T + p^{5} T^{2} \) |
| 47 | \( 1 + 19680 T + p^{5} T^{2} \) |
| 53 | \( 1 - 31266 T + p^{5} T^{2} \) |
| 59 | \( 1 - 26340 T + p^{5} T^{2} \) |
| 61 | \( 1 + 31090 T + p^{5} T^{2} \) |
| 67 | \( 1 - 16804 T + p^{5} T^{2} \) |
| 71 | \( 1 - 6120 T + p^{5} T^{2} \) |
| 73 | \( 1 + 25558 T + p^{5} T^{2} \) |
| 79 | \( 1 + 74408 T + p^{5} T^{2} \) |
| 83 | \( 1 + 6468 T + p^{5} T^{2} \) |
| 89 | \( 1 - 32742 T + p^{5} T^{2} \) |
| 97 | \( 1 - 166082 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.82779050664734778527052649632, −10.15489081859172396485728813413, −9.896545068752947595459228635863, −8.760809989637413458372694478808, −7.13119755837099804734403659738, −6.20997823447514064629569932264, −5.12418426347401793692091307834, −3.30742614915159950285938965112, −2.01619511103667959607515431165, 0,
2.01619511103667959607515431165, 3.30742614915159950285938965112, 5.12418426347401793692091307834, 6.20997823447514064629569932264, 7.13119755837099804734403659738, 8.760809989637413458372694478808, 9.896545068752947595459228635863, 10.15489081859172396485728813413, 11.82779050664734778527052649632
