| L(s) = 1 | − 2.64i·7-s + 3·9-s − 5.29·11-s − 5.29i·23-s − 5·25-s − 2i·29-s − 6i·37-s − 5.29·43-s − 7.00·49-s + 10i·53-s − 7.93i·63-s − 15.8·67-s − 5.29i·71-s + 14.0i·77-s − 15.8i·79-s + ⋯ |
| L(s) = 1 | − 0.999i·7-s + 9-s − 1.59·11-s − 1.10i·23-s − 25-s − 0.371i·29-s − 0.986i·37-s − 0.806·43-s − 49-s + 1.37i·53-s − 0.999i·63-s − 1.93·67-s − 0.627i·71-s + 1.59i·77-s − 1.78i·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9148831122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9148831122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 5.29iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 5.29iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.026115575831226771600767065285, −7.87519131557572434521216330137, −7.59649726418208710355607065791, −6.71497343406342298249967705676, −5.74823144586505082747215942590, −4.70751578525888815984967594473, −4.12250871164303618833151258929, −2.96267474376401946616515117236, −1.79450059897825166281683950311, −0.32399554543082494128334546995,
1.64269119880297867522777089324, 2.62643578659415636288269685106, 3.64346788639910804449702837756, 4.87810160201851801591789898723, 5.41043382503614681747357274451, 6.33246660329563653551846667278, 7.35094387548997427736519404134, 7.966006442695055636142877899678, 8.729216301003854646069576725067, 9.775757146573598558236201969928
