| L(s) = 1 | − 2-s − 3-s + 4-s + 2.56·5-s + 6-s − 8-s + 9-s − 2.56·10-s + 3·11-s − 12-s + 3.12·13-s − 2.56·15-s + 16-s + 4.68·17-s − 18-s + 3.12·19-s + 2.56·20-s − 3·22-s − 23-s + 24-s + 1.56·25-s − 3.12·26-s − 27-s + 0.123·29-s + 2.56·30-s + 1.68·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.14·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.810·10-s + 0.904·11-s − 0.288·12-s + 0.866·13-s − 0.661·15-s + 0.250·16-s + 1.13·17-s − 0.235·18-s + 0.716·19-s + 0.572·20-s − 0.639·22-s − 0.208·23-s + 0.204·24-s + 0.312·25-s − 0.612·26-s − 0.192·27-s + 0.0228·29-s + 0.467·30-s + 0.302·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.951135287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.951135287\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 29 | \( 1 - 0.123T + 29T^{2} \) |
| 31 | \( 1 - 1.68T + 31T^{2} \) |
| 37 | \( 1 - 0.876T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 0.438T + 47T^{2} \) |
| 53 | \( 1 + 0.561T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 0.438T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 6.56T + 83T^{2} \) |
| 89 | \( 1 + 8.24T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.946159228754029699667088160716, −7.28199452400969424951747682143, −6.48137445335870395423981060305, −5.84755281523622648776351409902, −5.59547819018845606530905868379, −4.40206448072957935548303817768, −3.51576905874830444891817298948, −2.51513475398525113205969002379, −1.46327466353418879761095644705, −0.940607411340999308277410648802,
0.940607411340999308277410648802, 1.46327466353418879761095644705, 2.51513475398525113205969002379, 3.51576905874830444891817298948, 4.40206448072957935548303817768, 5.59547819018845606530905868379, 5.84755281523622648776351409902, 6.48137445335870395423981060305, 7.28199452400969424951747682143, 7.946159228754029699667088160716
