L(s) = 1 | + 2.74·3-s + 5-s + 1.90·7-s + 4.51·9-s + 3.61·11-s − 1.41·13-s + 2.74·15-s + 1.85·17-s − 0.985·19-s + 5.21·21-s + 3.95·23-s + 25-s + 4.16·27-s − 1.43·29-s + 7.26·31-s + 9.90·33-s + 1.90·35-s − 6.05·37-s − 3.86·39-s + 3.40·41-s + 0.326·43-s + 4.51·45-s + 8.86·47-s − 3.38·49-s + 5.07·51-s − 7.15·53-s + 3.61·55-s + ⋯ |
L(s) = 1 | + 1.58·3-s + 0.447·5-s + 0.718·7-s + 1.50·9-s + 1.08·11-s − 0.391·13-s + 0.707·15-s + 0.448·17-s − 0.226·19-s + 1.13·21-s + 0.825·23-s + 0.200·25-s + 0.800·27-s − 0.265·29-s + 1.30·31-s + 1.72·33-s + 0.321·35-s − 0.995·37-s − 0.619·39-s + 0.531·41-s + 0.0498·43-s + 0.673·45-s + 1.29·47-s − 0.483·49-s + 0.710·51-s − 0.982·53-s + 0.487·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.223632750\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.223632750\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 7 | \( 1 - 1.90T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 + 0.985T + 19T^{2} \) |
| 23 | \( 1 - 3.95T + 23T^{2} \) |
| 29 | \( 1 + 1.43T + 29T^{2} \) |
| 31 | \( 1 - 7.26T + 31T^{2} \) |
| 37 | \( 1 + 6.05T + 37T^{2} \) |
| 41 | \( 1 - 3.40T + 41T^{2} \) |
| 43 | \( 1 - 0.326T + 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 + 7.15T + 53T^{2} \) |
| 59 | \( 1 - 7.97T + 59T^{2} \) |
| 61 | \( 1 + 6.30T + 61T^{2} \) |
| 67 | \( 1 + 5.49T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 0.688T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 5.13T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 7.24T + 97T^{2} \) |
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
−7.892059407683379706527094993604, −7.32804384132406663031195554281, −6.64301405544849075483037864796, −5.77431387413471255961024436152, −4.80798919847944295432583070618, −4.20863921782212346571242540811, −3.36444103783661344445780515773, −2.69091044463582228140142974062, −1.86002193180096921113586406786, −1.16526583801642066356722894145,
1.16526583801642066356722894145, 1.86002193180096921113586406786, 2.69091044463582228140142974062, 3.36444103783661344445780515773, 4.20863921782212346571242540811, 4.80798919847944295432583070618, 5.77431387413471255961024436152, 6.64301405544849075483037864796, 7.32804384132406663031195554281, 7.892059407683379706527094993604