| L(s) = 1 | + 2.68·2-s + 0.795·3-s + 5.20·4-s + 1.19·5-s + 2.13·6-s − 2.78·7-s + 8.60·8-s − 2.36·9-s + 3.21·10-s − 1.31·11-s + 4.13·12-s − 1.85·13-s − 7.47·14-s + 0.950·15-s + 12.6·16-s − 3.80·17-s − 6.35·18-s + 3.71·19-s + 6.22·20-s − 2.21·21-s − 3.53·22-s + 1.17·23-s + 6.84·24-s − 3.56·25-s − 4.97·26-s − 4.26·27-s − 14.4·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.459·3-s + 2.60·4-s + 0.534·5-s + 0.871·6-s − 1.05·7-s + 3.04·8-s − 0.789·9-s + 1.01·10-s − 0.396·11-s + 1.19·12-s − 0.513·13-s − 1.99·14-s + 0.245·15-s + 3.17·16-s − 0.923·17-s − 1.49·18-s + 0.851·19-s + 1.39·20-s − 0.482·21-s − 0.752·22-s + 0.244·23-s + 1.39·24-s − 0.713·25-s − 0.975·26-s − 0.821·27-s − 2.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.614865066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.614865066\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 - T \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 3 | \( 1 - 0.795T + 3T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 7 | \( 1 + 2.78T + 7T^{2} \) |
| 11 | \( 1 + 1.31T + 11T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 - 3.71T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 - 0.692T + 29T^{2} \) |
| 31 | \( 1 - 2.81T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 - 7.41T + 41T^{2} \) |
| 43 | \( 1 - 3.18T + 43T^{2} \) |
| 47 | \( 1 - 4.31T + 47T^{2} \) |
| 53 | \( 1 - 5.88T + 53T^{2} \) |
| 59 | \( 1 + 6.08T + 59T^{2} \) |
| 61 | \( 1 - 7.30T + 61T^{2} \) |
| 67 | \( 1 - 4.90T + 67T^{2} \) |
| 71 | \( 1 + 8.88T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 3.50T + 83T^{2} \) |
| 89 | \( 1 - 0.772T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−11.11613357480619832709352135720, −10.11512223029604164387733897041, −9.159240615178719104774311031982, −7.74831122782822809588562252587, −6.78722785057364672029956038799, −5.95648036046349874599340369415, −5.25231842504491230477835130075, −4.03156870882850897287791210129, −2.97417232967726473325539190877, −2.34357699635306637131244904045,
2.34357699635306637131244904045, 2.97417232967726473325539190877, 4.03156870882850897287791210129, 5.25231842504491230477835130075, 5.95648036046349874599340369415, 6.78722785057364672029956038799, 7.74831122782822809588562252587, 9.159240615178719104774311031982, 10.11512223029604164387733897041, 11.11613357480619832709352135720
