L(s) = 1 | − 0.810·2-s + 2.88·3-s − 1.34·4-s − 2.33·6-s + 3.73·7-s + 2.70·8-s + 5.34·9-s + 0.363·11-s − 3.88·12-s + 1.90·13-s − 3.02·14-s + 0.492·16-s + 1.56·17-s − 4.32·18-s − 4.74·19-s + 10.7·21-s − 0.294·22-s − 0.210·23-s + 7.82·24-s − 1.54·26-s + 6.76·27-s − 5.01·28-s + 3.72·29-s + 2.95·31-s − 5.81·32-s + 1.05·33-s − 1.27·34-s + ⋯ |
L(s) = 1 | − 0.572·2-s + 1.66·3-s − 0.671·4-s − 0.955·6-s + 1.41·7-s + 0.957·8-s + 1.78·9-s + 0.109·11-s − 1.12·12-s + 0.529·13-s − 0.808·14-s + 0.123·16-s + 0.380·17-s − 1.02·18-s − 1.08·19-s + 2.35·21-s − 0.0628·22-s − 0.0439·23-s + 1.59·24-s − 0.303·26-s + 1.30·27-s − 0.948·28-s + 0.691·29-s + 0.530·31-s − 1.02·32-s + 0.182·33-s − 0.217·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.125042089\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.125042089\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 0.810T + 2T^{2} \) |
| 3 | \( 1 - 2.88T + 3T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 - 0.363T + 11T^{2} \) |
| 13 | \( 1 - 1.90T + 13T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 + 0.210T + 23T^{2} \) |
| 29 | \( 1 - 3.72T + 29T^{2} \) |
| 31 | \( 1 - 2.95T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 5.79T + 43T^{2} \) |
| 47 | \( 1 + 3.70T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 + 5.13T + 59T^{2} \) |
| 61 | \( 1 - 3.62T + 61T^{2} \) |
| 67 | \( 1 + 1.38T + 67T^{2} \) |
| 71 | \( 1 + 0.880T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 + 0.286T + 79T^{2} \) |
| 83 | \( 1 - 6.19T + 83T^{2} \) |
| 89 | \( 1 + 2.54T + 89T^{2} \) |
| 97 | \( 1 + 8.23T + 97T^{2} \) |
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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
−8.214111017526533456931606494915, −7.83301349136588941746458631634, −7.10426297553845508689548453901, −5.97400055067253829904881661135, −4.86449529974524672008338777424, −4.32341012039576309621641571762, −3.70966239463953820986687659347, −2.62676880094790247928182435604, −1.80146569363211566788881294825, −1.03459520022394156215802882991,
1.03459520022394156215802882991, 1.80146569363211566788881294825, 2.62676880094790247928182435604, 3.70966239463953820986687659347, 4.32341012039576309621641571762, 4.86449529974524672008338777424, 5.97400055067253829904881661135, 7.10426297553845508689548453901, 7.83301349136588941746458631634, 8.214111017526533456931606494915