L(s) = 1 | + 2i·3-s − 3.46·5-s + (2 − 1.73i)7-s − 9-s + 3.46·11-s − 3.46·13-s − 6.92i·15-s + 2i·19-s + (3.46 + 4i)21-s + 3.46i·23-s + 6.99·25-s + 4i·27-s + 6i·29-s − 8·31-s + 6.92i·33-s + ⋯ |
L(s) = 1 | + 1.15i·3-s − 1.54·5-s + (0.755 − 0.654i)7-s − 0.333·9-s + 1.04·11-s − 0.960·13-s − 1.78i·15-s + 0.458i·19-s + (0.755 + 0.872i)21-s + 0.722i·23-s + 1.39·25-s + 0.769i·27-s + 1.11i·29-s − 1.43·31-s + 1.20i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6798122148\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6798122148\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−9.663645281554354836364521315266, −8.946502052394659249505128593852, −8.043980990229914926428917012657, −7.42832358741787704815870031138, −6.71696830135905326800491183742, −5.11870142618711833205801656335, −4.68752005536493134425437009207, −3.75542677648097625614275171449, −3.47310751387342768724261869625, −1.48779104086079354878900818507,
0.26578833208435336949568569617, 1.61910428636712216156771060209, 2.65584026475610212031884129966, 3.96038811704835909635138117478, 4.63469641277580437120517900699, 5.76040004110084872342578564834, 6.87661213633771044322547220647, 7.29003369722086081256045204600, 8.009595417901541059394552839332, 8.613650944040973539157450600256