L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 1.23·7-s + 8-s + 9-s + 2·11-s − 12-s + 5.23·13-s − 1.23·14-s + 16-s + 17-s + 18-s − 6.47·19-s + 1.23·21-s + 2·22-s + 4·23-s − 24-s + 5.23·26-s − 27-s − 1.23·28-s − 4·29-s + 2.47·31-s + 32-s − 2·33-s + 34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.467·7-s + 0.353·8-s + 0.333·9-s + 0.603·11-s − 0.288·12-s + 1.45·13-s − 0.330·14-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 1.48·19-s + 0.269·21-s + 0.426·22-s + 0.834·23-s − 0.204·24-s + 1.02·26-s − 0.192·27-s − 0.233·28-s − 0.742·29-s + 0.444·31-s + 0.176·32-s − 0.348·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.454358319\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.454358319\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 1.70T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 8.76T + 73T^{2} \) |
| 79 | \( 1 - 0.944T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 1.70T + 97T^{2} \) |
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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
−8.904775858294584676725791637040, −8.101452541666930245263362201127, −7.03640486158265976828233047745, −6.37088077843960488701549741047, −5.94908397054616690306763631223, −4.97363440998996522187272083336, −4.03468078988914248637004972596, −3.48180097378600502598923141681, −2.18368921204496976796195123769, −0.953090192913917929664689264022,
0.953090192913917929664689264022, 2.18368921204496976796195123769, 3.48180097378600502598923141681, 4.03468078988914248637004972596, 4.97363440998996522187272083336, 5.94908397054616690306763631223, 6.37088077843960488701549741047, 7.03640486158265976828233047745, 8.101452541666930245263362201127, 8.904775858294584676725791637040