L(s) = 1 | + 2-s + 3-s + 4-s + 0.267·5-s + 6-s + 0.732·7-s + 8-s + 9-s + 0.267·10-s + 4.73·11-s + 12-s + 0.732·14-s + 0.267·15-s + 16-s − 2.26·17-s + 18-s + 1.26·19-s + 0.267·20-s + 0.732·21-s + 4.73·22-s − 6.19·23-s + 24-s − 4.92·25-s + 27-s + 0.732·28-s + 2.46·29-s + 0.267·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.119·5-s + 0.408·6-s + 0.276·7-s + 0.353·8-s + 0.333·9-s + 0.0847·10-s + 1.42·11-s + 0.288·12-s + 0.195·14-s + 0.0691·15-s + 0.250·16-s − 0.550·17-s + 0.235·18-s + 0.290·19-s + 0.0599·20-s + 0.159·21-s + 1.00·22-s − 1.29·23-s + 0.204·24-s − 0.985·25-s + 0.192·27-s + 0.138·28-s + 0.457·29-s + 0.0489·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.328312941\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.328312941\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 0.267T + 5T^{2} \) |
| 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 + 6.19T + 23T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 7.66T + 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 - 0.464T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 + 9.73T + 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.774399464370172005642364672142, −9.257963737451540274950828945425, −8.191260315817174753599323194746, −7.43515661949823690334986904425, −6.43010019651704500625992494719, −5.74591359033463877894660637024, −4.35013220867184309438563082123, −3.91784500765718652165560949092, −2.60995926320768441118217032089, −1.52147856856187922875477354307,
1.52147856856187922875477354307, 2.60995926320768441118217032089, 3.91784500765718652165560949092, 4.35013220867184309438563082123, 5.74591359033463877894660637024, 6.43010019651704500625992494719, 7.43515661949823690334986904425, 8.191260315817174753599323194746, 9.257963737451540274950828945425, 9.774399464370172005642364672142