| L(s) = 1 | − 2-s + 0.0255·3-s + 4-s − 1.15·5-s − 0.0255·6-s − 2.71·7-s − 8-s − 2.99·9-s + 1.15·10-s + 2.25·11-s + 0.0255·12-s + 2.71·14-s − 0.0293·15-s + 16-s − 1.05·17-s + 2.99·18-s + 19-s − 1.15·20-s − 0.0692·21-s − 2.25·22-s − 4.19·23-s − 0.0255·24-s − 3.67·25-s − 0.153·27-s − 2.71·28-s − 9.59·29-s + 0.0293·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.0147·3-s + 0.5·4-s − 0.514·5-s − 0.0104·6-s − 1.02·7-s − 0.353·8-s − 0.999·9-s + 0.363·10-s + 0.679·11-s + 0.00736·12-s + 0.725·14-s − 0.00757·15-s + 0.250·16-s − 0.255·17-s + 0.706·18-s + 0.229·19-s − 0.257·20-s − 0.0151·21-s − 0.480·22-s − 0.874·23-s − 0.00520·24-s − 0.735·25-s − 0.0294·27-s − 0.513·28-s − 1.78·29-s + 0.00535·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4622485306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4622485306\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0255T + 3T^{2} \) |
| 5 | \( 1 + 1.15T + 5T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 - 2.25T + 11T^{2} \) |
| 17 | \( 1 + 1.05T + 17T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 + 9.59T + 29T^{2} \) |
| 31 | \( 1 - 5.93T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 + 4.42T + 43T^{2} \) |
| 47 | \( 1 - 1.18T + 47T^{2} \) |
| 53 | \( 1 + 2.39T + 53T^{2} \) |
| 59 | \( 1 + 2.95T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 - 4.39T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 0.865T + 73T^{2} \) |
| 79 | \( 1 + 1.38T + 79T^{2} \) |
| 83 | \( 1 - 7.32T + 83T^{2} \) |
| 89 | \( 1 - 8.81T + 89T^{2} \) |
| 97 | \( 1 + 3.14T + 97T^{2} \) |
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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
−8.000230993041622266492874078475, −7.51741536543559231597553572659, −6.52690379209425738132750991880, −6.18030611980158552652137006376, −5.38234063955013043899484976533, −4.17328882383951032014182603847, −3.50125425365893850517867809740, −2.77441371684312869407433972400, −1.75651123707194871249523126094, −0.37652529803565193535058129365,
0.37652529803565193535058129365, 1.75651123707194871249523126094, 2.77441371684312869407433972400, 3.50125425365893850517867809740, 4.17328882383951032014182603847, 5.38234063955013043899484976533, 6.18030611980158552652137006376, 6.52690379209425738132750991880, 7.51741536543559231597553572659, 8.000230993041622266492874078475
