L(s) = 1 | − 0.449·5-s − 2.04·7-s − 3.46·11-s − 4.79·25-s + 9.34·29-s − 3.60·31-s + 0.921·35-s − 2.79·49-s + 7.55·53-s + 1.55·55-s − 11.3·59-s + 9.79·73-s + 7.10·77-s + 17.4·79-s + 17.3·83-s − 2·97-s + 16.4·101-s + 0.492·103-s − 11.3·107-s + ⋯ |
L(s) = 1 | − 0.201·5-s − 0.774·7-s − 1.04·11-s − 0.959·25-s + 1.73·29-s − 0.647·31-s + 0.155·35-s − 0.399·49-s + 1.03·53-s + 0.209·55-s − 1.47·59-s + 1.14·73-s + 0.809·77-s + 1.96·79-s + 1.90·83-s − 0.203·97-s + 1.63·101-s + 0.0485·103-s − 1.09·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.171575595\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.171575595\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.449T + 5T^{2} \) |
| 7 | \( 1 + 2.04T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.34T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 7.55T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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
−8.144633369341423834853447350098, −7.73756911161141834611494618007, −6.80912890772934283225729495968, −6.20656166274815553886594958606, −5.37296693200450229726460147129, −4.63961203850706955314039903208, −3.66949506639917000478347621673, −2.94268475704182839477958847624, −2.05838866376346706104409436290, −0.57710219870707966830547024501,
0.57710219870707966830547024501, 2.05838866376346706104409436290, 2.94268475704182839477958847624, 3.66949506639917000478347621673, 4.63961203850706955314039903208, 5.37296693200450229726460147129, 6.20656166274815553886594958606, 6.80912890772934283225729495968, 7.73756911161141834611494618007, 8.144633369341423834853447350098