| L(s) = 1 | + 0.517i·2-s + 1.73·4-s + (1.73 − 1.41i)5-s + 2.44i·7-s + 1.93i·8-s + (0.732 + 0.896i)10-s + 1.26·11-s − 1.79i·13-s − 1.26·14-s + 2.46·16-s + 1.41i·17-s + 3.26·19-s + (3 − 2.44i)20-s + 0.656i·22-s + 6.31i·23-s + ⋯ |
| L(s) = 1 | + 0.366i·2-s + 0.866·4-s + (0.774 − 0.632i)5-s + 0.925i·7-s + 0.683i·8-s + (0.231 + 0.283i)10-s + 0.382·11-s − 0.497i·13-s − 0.338·14-s + 0.616·16-s + 0.342i·17-s + 0.749·19-s + (0.670 − 0.547i)20-s + 0.139i·22-s + 1.31i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.501483477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.501483477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.73 + 1.41i)T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.517iT - 2T^{2} \) |
| 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 + 1.79iT - 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 - 6.31iT - 23T^{2} \) |
| 31 | \( 1 + 8.73T + 31T^{2} \) |
| 37 | \( 1 + 9.14iT - 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 9.14iT - 43T^{2} \) |
| 47 | \( 1 - 1.41iT - 47T^{2} \) |
| 53 | \( 1 + 5.93iT - 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 7.34iT - 73T^{2} \) |
| 79 | \( 1 - 4.19T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 10.9iT - 97T^{2} \) |
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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
−9.434585582807516998746029973277, −9.099112680037768184376398674345, −7.993815731179716789995958250124, −7.35643798380364351062798965399, −6.15672906155048793008843565666, −5.74185140181659740906634609545, −5.03275282368382959792285631489, −3.49134581590667326319321262017, −2.36804620841260282479037982889, −1.45732607131558302754149466085,
1.18459633480461727541049630143, 2.28253277097415837472278729178, 3.22672639264062564852996615128, 4.19895240962364984137106134266, 5.50092530078822640510322272217, 6.47472115167800355813335429708, 6.98897406328302919769011621327, 7.64627728506605889852178002707, 8.985577533557513645386761872307, 9.774076061328306375135937992099
