| L(s) = 1 | + 2.99·3-s − 4.21·7-s + 5.97·9-s + 2.94·11-s − 3.21·13-s + 5.01·17-s + 4.17·19-s − 12.6·21-s − 2.12·23-s + 8.92·27-s + 7.10·29-s + 1.13·31-s + 8.83·33-s + 2.04·37-s − 9.62·39-s − 1.04·41-s − 3.45·43-s − 6.70·47-s + 10.7·49-s + 15.0·51-s − 3.16·53-s + 12.5·57-s + 9.30·59-s + 1.79·61-s − 25.1·63-s − 8.15·67-s − 6.38·69-s + ⋯ |
| L(s) = 1 | + 1.72·3-s − 1.59·7-s + 1.99·9-s + 0.888·11-s − 0.891·13-s + 1.21·17-s + 0.958·19-s − 2.75·21-s − 0.444·23-s + 1.71·27-s + 1.31·29-s + 0.204·31-s + 1.53·33-s + 0.335·37-s − 1.54·39-s − 0.163·41-s − 0.526·43-s − 0.977·47-s + 1.53·49-s + 2.10·51-s − 0.434·53-s + 1.65·57-s + 1.21·59-s + 0.230·61-s − 3.17·63-s − 0.996·67-s − 0.768·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.794946403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.794946403\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.99T + 3T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 13 | \( 1 + 3.21T + 13T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 + 2.12T + 23T^{2} \) |
| 29 | \( 1 - 7.10T + 29T^{2} \) |
| 31 | \( 1 - 1.13T + 31T^{2} \) |
| 37 | \( 1 - 2.04T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 59 | \( 1 - 9.30T + 59T^{2} \) |
| 61 | \( 1 - 1.79T + 61T^{2} \) |
| 67 | \( 1 + 8.15T + 67T^{2} \) |
| 71 | \( 1 - 8.85T + 71T^{2} \) |
| 73 | \( 1 + 1.06T + 73T^{2} \) |
| 79 | \( 1 - 7.68T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 5.45T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
| 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
−7.76023177438078735568364864676, −7.00833224998672428533494566686, −6.60138606979450961292757663377, −5.69595254258535811392187563450, −4.67432521179844303193444954117, −3.84260059761494126073488694570, −3.19149587946261602643729406353, −2.93335499804145523026412619866, −1.92956168398608773584555508002, −0.851625260925886615488620034401,
0.851625260925886615488620034401, 1.92956168398608773584555508002, 2.93335499804145523026412619866, 3.19149587946261602643729406353, 3.84260059761494126073488694570, 4.67432521179844303193444954117, 5.69595254258535811392187563450, 6.60138606979450961292757663377, 7.00833224998672428533494566686, 7.76023177438078735568364864676
