| L(s) = 1 | − 3·9-s + 2·11-s − 4·13-s + 2·17-s − 4·19-s + 4·23-s − 5·25-s + 6·29-s − 8·31-s − 2·37-s + 2·41-s + 10·43-s + 2·53-s + 8·59-s − 8·61-s + 2·67-s + 14·73-s + 4·79-s + 9·81-s − 12·83-s + 6·89-s − 6·97-s − 6·99-s + 16·101-s − 8·103-s − 2·107-s + 14·109-s + ⋯ |
| L(s) = 1 | − 9-s + 0.603·11-s − 1.10·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s − 25-s + 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.312·41-s + 1.52·43-s + 0.274·53-s + 1.04·59-s − 1.02·61-s + 0.244·67-s + 1.63·73-s + 0.450·79-s + 81-s − 1.31·83-s + 0.635·89-s − 0.609·97-s − 0.603·99-s + 1.59·101-s − 0.788·103-s − 0.193·107-s + 1.34·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.436455452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436455452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.023528955534108624166036868061, −7.35890903835713693504750392315, −6.64171591586812206488188905313, −5.86679534602402891862086481902, −5.26485144018969168063909036405, −4.42417243926141061580931897932, −3.60409221784191514018426306909, −2.72246755506876279735470460441, −1.96852077522635108794107034779, −0.60345679758312762559695317151,
0.60345679758312762559695317151, 1.96852077522635108794107034779, 2.72246755506876279735470460441, 3.60409221784191514018426306909, 4.42417243926141061580931897932, 5.26485144018969168063909036405, 5.86679534602402891862086481902, 6.64171591586812206488188905313, 7.35890903835713693504750392315, 8.023528955534108624166036868061
