| L(s) = 1 | + 1.41i·7-s + (−1 − i)13-s − 6·17-s + (4.24 + 4.24i)19-s − 8.48i·23-s + 5i·25-s + (6 + 6i)29-s + 1.41·31-s + (−5 + 5i)37-s + 6i·41-s + (−4.24 + 4.24i)43-s − 8.48·47-s + 5·49-s + (−6 + 6i)53-s + (−5 − 5i)61-s + ⋯ |
| L(s) = 1 | + 0.534i·7-s + (−0.277 − 0.277i)13-s − 1.45·17-s + (0.973 + 0.973i)19-s − 1.76i·23-s + i·25-s + (1.11 + 1.11i)29-s + 0.254·31-s + (−0.821 + 0.821i)37-s + 0.937i·41-s + (−0.646 + 0.646i)43-s − 1.23·47-s + 0.714·49-s + (−0.824 + 0.824i)53-s + (−0.640 − 0.640i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.077239594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.077239594\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 5iT^{2} \) |
| 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 - 11iT^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + (-4.24 - 4.24i)T + 19iT^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 + (-6 - 6i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (4.24 - 4.24i)T - 43iT^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + (6 - 6i)T - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (5 + 5i)T + 61iT^{2} \) |
| 67 | \( 1 + (-2.82 - 2.82i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
| 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
−9.166621301000003124149174251325, −8.489707121229663193461656375091, −7.84645658408280016024313893055, −6.74482403360509508164141066691, −6.30503206418632232052932633575, −5.13905627158435542365683695826, −4.64499638070020071096314864984, −3.37629555557710574972288974723, −2.58104023488496124966257506820, −1.37926782333916385170133901781,
0.36781669003870647882598076445, 1.82985133504901376573277304707, 2.88665714932374340011344779327, 3.95841575990851241331003382181, 4.70685389115854170088387727069, 5.55298413978185410955867840963, 6.64897060806520900504018887011, 7.10517181703890070943609204705, 7.974707149727411843867717692784, 8.813951607911538149889379514953
