L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 2.60·11-s + 12-s − 6.60·13-s − 14-s − 15-s + 16-s − 18-s + 2·19-s − 20-s + 21-s − 2.60·22-s − 23-s − 24-s + 25-s + 6.60·26-s + 27-s + 28-s − 8.60·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.785·11-s + 0.288·12-s − 1.83·13-s − 0.267·14-s − 0.258·15-s + 0.250·16-s − 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.218·21-s − 0.555·22-s − 0.208·23-s − 0.204·24-s + 0.200·25-s + 1.29·26-s + 0.192·27-s + 0.188·28-s − 1.59·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 + 6.60T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 6.60T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 6.60T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 9.39T + 89T^{2} \) |
| 97 | \( 1 - 3.81T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77756837227707404026828651061, −7.45424438370500421754538347502, −6.84076949991818514086097163118, −5.79615680334637179004851549456, −4.86595463845434335351446719509, −4.10513591724333639852740010707, −3.16078995066942221782912298014, −2.30650728528818035247321579076, −1.40494748041946689107416747220, 0,
1.40494748041946689107416747220, 2.30650728528818035247321579076, 3.16078995066942221782912298014, 4.10513591724333639852740010707, 4.86595463845434335351446719509, 5.79615680334637179004851549456, 6.84076949991818514086097163118, 7.45424438370500421754538347502, 7.77756837227707404026828651061