| L(s) = 1 | − i·3-s + 3i·7-s − 9-s − 11-s − 4i·13-s − i·17-s + 7·19-s + 3·21-s − 3i·23-s + i·27-s − 10·29-s + i·33-s + i·37-s − 4·39-s + 5·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.13i·7-s − 0.333·9-s − 0.301·11-s − 1.10i·13-s − 0.242i·17-s + 1.60·19-s + 0.654·21-s − 0.625i·23-s + 0.192i·27-s − 1.85·29-s + 0.174i·33-s + 0.164i·37-s − 0.640·39-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.624371073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.624371073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - iT - 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 5iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 - 15T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 13iT - 97T^{2} \) |
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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
−8.351161572594121621770995806641, −7.79853084098359768469355396924, −7.14749451606471135120713404210, −6.10267587754654439196416387292, −5.52736443916960795743779627513, −4.97527217040294227871397351859, −3.52590412602665731135923148457, −2.80287065736740764426006422195, −1.93878776845974789058476352015, −0.59272747760092981467884833993,
1.00258537576644659561909555868, 2.22653773169493018400789568809, 3.58695124873011118541631158809, 3.89794520825112203109109355050, 4.93321685893448072198701626601, 5.59009535741271586079673791281, 6.59891542821027260788639507218, 7.43996900952918732857542478708, 7.78811813176051148669011854464, 9.040248209804586867120837552701
