| L(s) = 1 | + 2-s + 4-s + 0.274i·5-s + 7-s + 8-s + 0.274i·10-s − 1.13i·11-s + 5.87i·13-s + 14-s + 16-s + 3.96i·17-s + (−4.35 + 0.274i)19-s + 0.274i·20-s − 1.13i·22-s − 1.63i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.122i·5-s + 0.377·7-s + 0.353·8-s + 0.0867i·10-s − 0.343i·11-s + 1.63i·13-s + 0.267·14-s + 0.250·16-s + 0.962i·17-s + (−0.998 + 0.0629i)19-s + 0.0613i·20-s − 0.243i·22-s − 0.340i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.802522017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.802522017\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + (4.35 - 0.274i)T \) |
| good | 5 | \( 1 - 0.274iT - 5T^{2} \) |
| 11 | \( 1 + 1.13iT - 11T^{2} \) |
| 13 | \( 1 - 5.87iT - 13T^{2} \) |
| 17 | \( 1 - 3.96iT - 17T^{2} \) |
| 23 | \( 1 + 1.63iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 7.51iT - 31T^{2} \) |
| 37 | \( 1 - 0.495iT - 37T^{2} \) |
| 41 | \( 1 - 4.31T + 41T^{2} \) |
| 43 | \( 1 + 8.31T + 43T^{2} \) |
| 47 | \( 1 - 4.73iT - 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 59 | \( 1 - 9.92T + 59T^{2} \) |
| 61 | \( 1 + 1.61T + 61T^{2} \) |
| 67 | \( 1 - 3.10iT - 67T^{2} \) |
| 71 | \( 1 + 9.14T + 71T^{2} \) |
| 73 | \( 1 - 5.53T + 73T^{2} \) |
| 79 | \( 1 + 6.42iT - 79T^{2} \) |
| 83 | \( 1 + 6.15iT - 83T^{2} \) |
| 89 | \( 1 - 8.31T + 89T^{2} \) |
| 97 | \( 1 + 3.36iT - 97T^{2} \) |
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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
−8.721349190958602696936850637240, −8.577605336397880101353446484459, −7.33434740941454749721518736959, −6.59329371357375698036891323389, −6.11485475745243857293264984405, −4.92731918176413635833845972500, −4.37672715697496425203925610968, −3.49899291157359139532199276730, −2.37919206352662369459055310857, −1.43701140788574600062218503628,
0.76290571255220824317944398720, 2.23796807805985323414699574990, 3.03965424053853612907338194816, 4.08175466235855509753302296484, 4.95542041952118871408030900294, 5.49725380023904459549051890051, 6.44534969980717046892598556131, 7.25879257435948378442158937480, 8.004350034271618792063013875691, 8.675159976973511134634289601261
