| L(s) = 1 | + i·3-s + (0.224 + 2.22i)5-s − 2i·7-s − 9-s + 0.449·11-s + i·13-s + (−2.22 + 0.224i)15-s + 2i·17-s + 6.89·19-s + 2·21-s − 4.89i·23-s + (−4.89 + i)25-s − i·27-s + 2·29-s + 2.89·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.100 + 0.994i)5-s − 0.755i·7-s − 0.333·9-s + 0.135·11-s + 0.277i·13-s + (−0.574 + 0.0580i)15-s + 0.485i·17-s + 1.58·19-s + 0.436·21-s − 1.02i·23-s + (−0.979 + 0.200i)25-s − 0.192i·27-s + 0.371·29-s + 0.520·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801131397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801131397\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.224 - 2.22i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 0.449T + 11T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 23 | \( 1 + 4.89iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 2.89T + 31T^{2} \) |
| 37 | \( 1 - 10.8iT - 37T^{2} \) |
| 41 | \( 1 - 3.55T + 41T^{2} \) |
| 43 | \( 1 - 7.79iT - 43T^{2} \) |
| 47 | \( 1 + 5.34iT - 47T^{2} \) |
| 53 | \( 1 - 2.89iT - 53T^{2} \) |
| 59 | \( 1 + 4.44T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 7.79iT - 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 7.79iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 8.44iT - 83T^{2} \) |
| 89 | \( 1 + 1.34T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.961879681283150504652634844964, −8.071774924502760771661133839708, −7.36663940513635388512388801454, −6.61895950687381683992209047757, −5.99045524116270797365540861354, −4.92224071957982395158452888152, −4.15739122884206301328219970749, −3.32272431879932632767879708477, −2.60147321404295083242761006832, −1.14710442091577692404417370568,
0.64181600834108421300333854403, 1.66316355125805489254068672149, 2.68446211774007186100054018362, 3.68469653693722745154837121901, 4.84084025883655593655450620913, 5.53234845833296926118454898014, 5.98120079727834080983431412515, 7.20229843498933223527432197503, 7.70852292287552622266676095373, 8.512681950782854349269963668454
