| L(s) = 1 | + 2.41i·3-s − 0.828i·7-s − 2.82·9-s + 5.24·11-s + 5.65i·13-s + 0.171i·17-s − 1.58·19-s + 1.99·21-s − 4.82i·23-s + 0.414i·27-s + 8·29-s − 0.828·31-s + 12.6i·33-s − 7.65i·37-s − 13.6·39-s + ⋯ |
| L(s) = 1 | + 1.39i·3-s − 0.313i·7-s − 0.942·9-s + 1.58·11-s + 1.56i·13-s + 0.0416i·17-s − 0.363·19-s + 0.436·21-s − 1.00i·23-s + 0.0797i·27-s + 1.48·29-s − 0.148·31-s + 2.20i·33-s − 1.25i·37-s − 2.18·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.030180990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.030180990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.41iT - 3T^{2} \) |
| 7 | \( 1 + 0.828iT - 7T^{2} \) |
| 11 | \( 1 - 5.24T + 11T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 0.171iT - 17T^{2} \) |
| 19 | \( 1 + 1.58T + 19T^{2} \) |
| 23 | \( 1 + 4.82iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 0.828T + 31T^{2} \) |
| 37 | \( 1 + 7.65iT - 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 9.65iT - 47T^{2} \) |
| 53 | \( 1 + 7.65iT - 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 2.75iT - 67T^{2} \) |
| 71 | \( 1 + 9.65T + 71T^{2} \) |
| 73 | \( 1 - 5.82iT - 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 1.24iT - 83T^{2} \) |
| 89 | \( 1 + 6.17T + 89T^{2} \) |
| 97 | \( 1 - 17.3iT - 97T^{2} \) |
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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
−9.089254489784495562641201598967, −8.530698730767911984812622322643, −7.31080280529188041001547449189, −6.54292995416076769419746816356, −5.95847788950885927387170638100, −4.53559768516190280538271923819, −4.40542877739915427289843707680, −3.72436716986378944806337876103, −2.54487397297577558504565877646, −1.22135110879854555129950286600,
0.72892854889154389567508267527, 1.52946829973955111247373568955, 2.59563396718800989567769125454, 3.48944405318447538843669889295, 4.57309971735887653706472596075, 5.78235976155411830652495521193, 6.12506060764495997377020046460, 7.07937674851210265228725979234, 7.48473828465752677657447730608, 8.451461370598083816614625729606
