L(s) = 1 | − 2.82·7-s − 3·9-s + 6.32·11-s + 4.47·13-s + 6.32·19-s − 8.48·23-s − 4.47·37-s + 2·41-s + 2.82·47-s + 1.00·49-s + 13.4·53-s − 6.32·59-s + 8.48·63-s − 17.8·77-s + 9·81-s + 14·89-s − 12.6·91-s − 18.9·99-s − 19.7·103-s − 13.4·117-s + ⋯ |
L(s) = 1 | − 1.06·7-s − 9-s + 1.90·11-s + 1.24·13-s + 1.45·19-s − 1.76·23-s − 0.735·37-s + 0.312·41-s + 0.412·47-s + 0.142·49-s + 1.84·53-s − 0.823·59-s + 1.06·63-s − 2.03·77-s + 81-s + 1.48·89-s − 1.32·91-s − 1.90·99-s − 1.95·103-s − 1.24·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.816187221\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.816187221\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 6.32T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 6.32T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 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
−8.134553372307698207702795944767, −7.20728290826308034806476580915, −6.42902627488124282578423824801, −6.06058322064431634580692511014, −5.42037356405753131005106594040, −4.07240999460776770777394491068, −3.65733158171779695015431377047, −2.97428307259247394288770954499, −1.74194202384615726561746861635, −0.71570579983271851545100075672,
0.71570579983271851545100075672, 1.74194202384615726561746861635, 2.97428307259247394288770954499, 3.65733158171779695015431377047, 4.07240999460776770777394491068, 5.42037356405753131005106594040, 6.06058322064431634580692511014, 6.42902627488124282578423824801, 7.20728290826308034806476580915, 8.134553372307698207702795944767