| L(s) = 1 | + 3-s − 3.12·5-s − 0.760·7-s + 9-s + 3.50·11-s − 3.64·13-s − 3.12·15-s + 6.89·17-s − 2.74·19-s − 0.760·21-s + 23-s + 4.73·25-s + 27-s − 29-s − 5.32·31-s + 3.50·33-s + 2.37·35-s + 5.93·37-s − 3.64·39-s + 6.52·41-s − 1.04·43-s − 3.12·45-s − 8.77·47-s − 6.42·49-s + 6.89·51-s + 6.18·53-s − 10.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.39·5-s − 0.287·7-s + 0.333·9-s + 1.05·11-s − 1.01·13-s − 0.805·15-s + 1.67·17-s − 0.629·19-s − 0.166·21-s + 0.208·23-s + 0.947·25-s + 0.192·27-s − 0.185·29-s − 0.955·31-s + 0.610·33-s + 0.401·35-s + 0.975·37-s − 0.584·39-s + 1.01·41-s − 0.158·43-s − 0.465·45-s − 1.28·47-s − 0.917·49-s + 0.965·51-s + 0.849·53-s − 1.47·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.668015349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.668015349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 3.12T + 5T^{2} \) |
| 7 | \( 1 + 0.760T + 7T^{2} \) |
| 11 | \( 1 - 3.50T + 11T^{2} \) |
| 13 | \( 1 + 3.64T + 13T^{2} \) |
| 17 | \( 1 - 6.89T + 17T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 - 5.93T + 37T^{2} \) |
| 41 | \( 1 - 6.52T + 41T^{2} \) |
| 43 | \( 1 + 1.04T + 43T^{2} \) |
| 47 | \( 1 + 8.77T + 47T^{2} \) |
| 53 | \( 1 - 6.18T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 0.576T + 61T^{2} \) |
| 67 | \( 1 - 3.98T + 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 - 7.28T + 73T^{2} \) |
| 79 | \( 1 - 6.45T + 79T^{2} \) |
| 83 | \( 1 - 2.58T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
| 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
−7.84664518345661065274140297973, −7.31793981635778601060951185212, −6.67243015135006518919027970485, −5.78453816490652889158964905129, −4.80258873514665009454739882651, −4.14226166182685167261323378297, −3.50943630416459516371677093386, −2.94383304722950697747091844021, −1.76372631882040785730639114767, −0.62398524502311777304020449595,
0.62398524502311777304020449595, 1.76372631882040785730639114767, 2.94383304722950697747091844021, 3.50943630416459516371677093386, 4.14226166182685167261323378297, 4.80258873514665009454739882651, 5.78453816490652889158964905129, 6.67243015135006518919027970485, 7.31793981635778601060951185212, 7.84664518345661065274140297973
