| L(s) = 1 | + 2-s + 3-s + 4-s − 3.71·5-s + 6-s − 7-s + 8-s + 9-s − 3.71·10-s + 6.59·11-s + 12-s − 13-s − 14-s − 3.71·15-s + 16-s + 17-s + 18-s + 0.160·19-s − 3.71·20-s − 21-s + 6.59·22-s + 3.45·23-s + 24-s + 8.81·25-s − 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.66·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.17·10-s + 1.98·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.959·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.0368·19-s − 0.831·20-s − 0.218·21-s + 1.40·22-s + 0.719·23-s + 0.204·24-s + 1.76·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.252122718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.252122718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3.71T + 5T^{2} \) |
| 11 | \( 1 - 6.59T + 11T^{2} \) |
| 19 | \( 1 - 0.160T + 19T^{2} \) |
| 23 | \( 1 - 3.45T + 23T^{2} \) |
| 29 | \( 1 - 4.05T + 29T^{2} \) |
| 31 | \( 1 + 9.59T + 31T^{2} \) |
| 37 | \( 1 - 3.48T + 37T^{2} \) |
| 41 | \( 1 - 0.0625T + 41T^{2} \) |
| 43 | \( 1 - 0.386T + 43T^{2} \) |
| 47 | \( 1 + 3.27T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 - 1.69T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 0.826T + 71T^{2} \) |
| 73 | \( 1 + 8.18T + 73T^{2} \) |
| 79 | \( 1 - 5.95T + 79T^{2} \) |
| 83 | \( 1 + 5.87T + 83T^{2} \) |
| 89 | \( 1 - 0.365T + 89T^{2} \) |
| 97 | \( 1 - 7.96T + 97T^{2} \) |
| 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
−7.57661927151116625494542238858, −6.99031355385831348617699434984, −6.59709392832713679207479379935, −5.61528379899391838275379888441, −4.51849373411462837998913439355, −4.21492949234639580065381762090, −3.40051869524900900242623624352, −3.15401835043088672885895022019, −1.81214507638896126352414772158, −0.77433277077740982403644981107,
0.77433277077740982403644981107, 1.81214507638896126352414772158, 3.15401835043088672885895022019, 3.40051869524900900242623624352, 4.21492949234639580065381762090, 4.51849373411462837998913439355, 5.61528379899391838275379888441, 6.59709392832713679207479379935, 6.99031355385831348617699434984, 7.57661927151116625494542238858
