| L(s) = 1 | + 0.786·3-s + 2.31·5-s − 1.15·7-s − 2.38·9-s − 4.35·11-s − 4.90·13-s + 1.81·15-s − 2.10·17-s + 0.609·19-s − 0.905·21-s + 7.29·23-s + 0.343·25-s − 4.23·27-s + 7.02·29-s − 0.954·31-s − 3.42·33-s − 2.66·35-s + 4.42·37-s − 3.86·39-s + 0.939·41-s + 2.08·43-s − 5.50·45-s + 3.32·47-s − 5.67·49-s − 1.65·51-s + 8.35·53-s − 10.0·55-s + ⋯ |
| L(s) = 1 | + 0.454·3-s + 1.03·5-s − 0.434·7-s − 0.793·9-s − 1.31·11-s − 1.36·13-s + 0.469·15-s − 0.509·17-s + 0.139·19-s − 0.197·21-s + 1.52·23-s + 0.0687·25-s − 0.814·27-s + 1.30·29-s − 0.171·31-s − 0.595·33-s − 0.449·35-s + 0.727·37-s − 0.618·39-s + 0.146·41-s + 0.318·43-s − 0.820·45-s + 0.485·47-s − 0.810·49-s − 0.231·51-s + 1.14·53-s − 1.35·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.923012900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923012900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 113 | \( 1 + T \) |
| good | 3 | \( 1 - 0.786T + 3T^{2} \) |
| 5 | \( 1 - 2.31T + 5T^{2} \) |
| 7 | \( 1 + 1.15T + 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 - 0.609T + 19T^{2} \) |
| 23 | \( 1 - 7.29T + 23T^{2} \) |
| 29 | \( 1 - 7.02T + 29T^{2} \) |
| 31 | \( 1 + 0.954T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 - 0.939T + 41T^{2} \) |
| 43 | \( 1 - 2.08T + 43T^{2} \) |
| 47 | \( 1 - 3.32T + 47T^{2} \) |
| 53 | \( 1 - 8.35T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 2.43T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 - 4.36T + 71T^{2} \) |
| 73 | \( 1 + 2.66T + 73T^{2} \) |
| 79 | \( 1 - 3.57T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 0.645T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.031999932460765928363912671976, −7.15601052468778228791942724818, −6.62121918753096164947691608060, −5.60058695516762965600720384925, −5.30341366352848737607084737783, −4.48441051576647202561243375199, −3.22549114242251305868366613050, −2.53895070623004224700030894766, −2.25602561188808175736064690936, −0.64387351668573119008028158054,
0.64387351668573119008028158054, 2.25602561188808175736064690936, 2.53895070623004224700030894766, 3.22549114242251305868366613050, 4.48441051576647202561243375199, 5.30341366352848737607084737783, 5.60058695516762965600720384925, 6.62121918753096164947691608060, 7.15601052468778228791942724818, 8.031999932460765928363912671976
