| L(s) = 1 | − 2-s + 0.331·3-s + 4-s + 2.59·5-s − 0.331·6-s + 4.59·7-s − 8-s − 2.89·9-s − 2.59·10-s + 4.86·11-s + 0.331·12-s − 1.10·13-s − 4.59·14-s + 0.860·15-s + 16-s − 2.33·17-s + 2.89·18-s + 3.93·19-s + 2.59·20-s + 1.52·21-s − 4.86·22-s − 5.22·23-s − 0.331·24-s + 1.75·25-s + 1.10·26-s − 1.95·27-s + 4.59·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.191·3-s + 0.5·4-s + 1.16·5-s − 0.135·6-s + 1.73·7-s − 0.353·8-s − 0.963·9-s − 0.821·10-s + 1.46·11-s + 0.0955·12-s − 0.305·13-s − 1.22·14-s + 0.222·15-s + 0.250·16-s − 0.566·17-s + 0.681·18-s + 0.902·19-s + 0.580·20-s + 0.331·21-s − 1.03·22-s − 1.08·23-s − 0.0675·24-s + 0.350·25-s + 0.216·26-s − 0.375·27-s + 0.868·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.378422585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.378422585\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 0.331T + 3T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 - 4.86T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 + 2.33T + 17T^{2} \) |
| 19 | \( 1 - 3.93T + 19T^{2} \) |
| 23 | \( 1 + 5.22T + 23T^{2} \) |
| 29 | \( 1 + 0.100T + 29T^{2} \) |
| 31 | \( 1 + 4.62T + 31T^{2} \) |
| 37 | \( 1 - 4.67T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 6.14T + 43T^{2} \) |
| 47 | \( 1 - 0.214T + 47T^{2} \) |
| 53 | \( 1 - 5.43T + 53T^{2} \) |
| 59 | \( 1 + 6.96T + 59T^{2} \) |
| 61 | \( 1 - 5.66T + 61T^{2} \) |
| 67 | \( 1 + 3.56T + 67T^{2} \) |
| 71 | \( 1 - 3.76T + 71T^{2} \) |
| 73 | \( 1 + 4.07T + 73T^{2} \) |
| 79 | \( 1 - 0.491T + 79T^{2} \) |
| 83 | \( 1 + 1.99T + 83T^{2} \) |
| 89 | \( 1 - 9.98T + 89T^{2} \) |
| 97 | \( 1 + 5.18T + 97T^{2} \) |
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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
−8.546575936914590791180586271933, −7.82458176259847369202803544635, −7.18993184695697206634605918130, −6.03726664346851139717326517254, −5.77097794200225026487732588105, −4.76046577858987751635911038449, −3.83832697378159064359186310018, −2.47210798105552688402132111705, −1.91650132550068580239764826607, −1.04603938886419254354482318067,
1.04603938886419254354482318067, 1.91650132550068580239764826607, 2.47210798105552688402132111705, 3.83832697378159064359186310018, 4.76046577858987751635911038449, 5.77097794200225026487732588105, 6.03726664346851139717326517254, 7.18993184695697206634605918130, 7.82458176259847369202803544635, 8.546575936914590791180586271933
