| L(s) = 1 | − 4·7-s − 13-s − 4·19-s − 5·25-s + 10·31-s + 2·37-s − 6·41-s − 10·43-s + 9·49-s − 6·53-s − 2·61-s − 2·67-s + 12·71-s + 10·73-s − 10·79-s + 12·83-s − 12·89-s + 4·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.277·13-s − 0.917·19-s − 25-s + 1.79·31-s + 0.328·37-s − 0.937·41-s − 1.52·43-s + 9/7·49-s − 0.824·53-s − 0.256·61-s − 0.244·67-s + 1.42·71-s + 1.17·73-s − 1.12·79-s + 1.31·83-s − 1.27·89-s + 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5276427191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5276427191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.97586114072533, −12.43697379392842, −12.13376615935058, −11.63833576561024, −11.08069911158896, −10.50852535336645, −10.02304985496578, −9.724310392741675, −9.398140856794822, −8.668799606237497, −8.224629552394421, −7.864192660291362, −7.061227651467838, −6.593746194285743, −6.431987595800313, −5.819247699032805, −5.261250779315024, −4.560339045928794, −4.180243424500079, −3.379844237299786, −3.190699658834966, −2.424748259855448, −1.944391861070055, −1.053704342642036, −0.2165193009633610,
0.2165193009633610, 1.053704342642036, 1.944391861070055, 2.424748259855448, 3.190699658834966, 3.379844237299786, 4.180243424500079, 4.560339045928794, 5.261250779315024, 5.819247699032805, 6.431987595800313, 6.593746194285743, 7.061227651467838, 7.864192660291362, 8.224629552394421, 8.668799606237497, 9.398140856794822, 9.724310392741675, 10.02304985496578, 10.50852535336645, 11.08069911158896, 11.63833576561024, 12.13376615935058, 12.43697379392842, 12.97586114072533
