| L(s) = 1 | − 2·5-s − 6.92i·7-s + 6.92i·11-s − 2·13-s − 10·17-s − 20.7i·19-s + 27.7i·23-s − 21·25-s − 26·29-s + 6.92i·31-s + 13.8i·35-s − 26·37-s − 58·41-s + 48.4i·43-s − 69.2i·47-s + ⋯ |
| L(s) = 1 | − 0.400·5-s − 0.989i·7-s + 0.629i·11-s − 0.153·13-s − 0.588·17-s − 1.09i·19-s + 1.20i·23-s − 0.839·25-s − 0.896·29-s + 0.223i·31-s + 0.395i·35-s − 0.702·37-s − 1.41·41-s + 1.12i·43-s − 1.47i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1831249258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1831249258\) |
| \(L(2)\) |
|
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 | \( 1 + 2T + 25T^{2} \) |
| 7 | \( 1 + 6.92iT - 49T^{2} \) |
| 11 | \( 1 - 6.92iT - 121T^{2} \) |
| 13 | \( 1 + 2T + 169T^{2} \) |
| 17 | \( 1 + 10T + 289T^{2} \) |
| 19 | \( 1 + 20.7iT - 361T^{2} \) |
| 23 | \( 1 - 27.7iT - 529T^{2} \) |
| 29 | \( 1 + 26T + 841T^{2} \) |
| 31 | \( 1 - 6.92iT - 961T^{2} \) |
| 37 | \( 1 + 26T + 1.36e3T^{2} \) |
| 41 | \( 1 + 58T + 1.68e3T^{2} \) |
| 43 | \( 1 - 48.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 69.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 74T + 2.80e3T^{2} \) |
| 59 | \( 1 + 90.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 26T + 3.72e3T^{2} \) |
| 67 | \( 1 + 6.92iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46T + 5.32e3T^{2} \) |
| 79 | \( 1 - 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 82T + 7.92e3T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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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
−10.03339038021942265155617093094, −9.341573384592258607710989130980, −8.186931484919404454133865768759, −7.30454726657398001805928434464, −6.72783537549393540043726355964, −5.27728783387667732534606325156, −4.32682990891179909219678171999, −3.37548664029518989069451793032, −1.77008093139877958591002900531, −0.06602336337095114662774609823,
1.93233315050610312982951861936, 3.19551640293675521952493593074, 4.34201172346643658386682509998, 5.56255503334711989418259179337, 6.28407772241014722268335405618, 7.50058990697695686600016205770, 8.425752793492289960442165289762, 9.026804562832237329270032206298, 10.11721894220532623320476267332, 11.00171488376096661069126238030
