L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 3.69·7-s − 8-s + 9-s − 10-s + 3.04·11-s + 12-s + 3.69·14-s + 15-s + 16-s − 6.85·17-s − 18-s − 0.911·19-s + 20-s − 3.69·21-s − 3.04·22-s − 0.356·23-s − 24-s + 25-s + 27-s − 3.69·28-s + 10.5·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.39·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.919·11-s + 0.288·12-s + 0.986·14-s + 0.258·15-s + 0.250·16-s − 1.66·17-s − 0.235·18-s − 0.209·19-s + 0.223·20-s − 0.805·21-s − 0.650·22-s − 0.0744·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s − 0.697·28-s + 1.95·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 - 3.04T + 11T^{2} \) |
| 17 | \( 1 + 6.85T + 17T^{2} \) |
| 19 | \( 1 + 0.911T + 19T^{2} \) |
| 23 | \( 1 + 0.356T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 - 0.899T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 9.43T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 + 8.54T + 59T^{2} \) |
| 61 | \( 1 - 1.55T + 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 - 4.13T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 9.62T + 79T^{2} \) |
| 83 | \( 1 - 9.86T + 83T^{2} \) |
| 89 | \( 1 - 6.41T + 89T^{2} \) |
| 97 | \( 1 + 6.97T + 97T^{2} \) |
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
−8.104691228153348516964408593951, −6.94645167301802726389892555438, −6.58766210149022839191045106014, −6.16420068737008692104377651347, −4.85178894836009238082230669338, −3.95395936504462640546926682589, −3.08469333501723261952830086569, −2.39376491153984294693839340596, −1.37618356645984093457242761861, 0,
1.37618356645984093457242761861, 2.39376491153984294693839340596, 3.08469333501723261952830086569, 3.95395936504462640546926682589, 4.85178894836009238082230669338, 6.16420068737008692104377651347, 6.58766210149022839191045106014, 6.94645167301802726389892555438, 8.104691228153348516964408593951