| L(s) = 1 | + 0.411·2-s + 0.231·3-s − 1.83·4-s + 0.0951·6-s − 3.56·7-s − 1.57·8-s − 2.94·9-s − 4.00·11-s − 0.423·12-s − 3.83·13-s − 1.46·14-s + 3.01·16-s − 0.709·17-s − 1.21·18-s − 6.25·19-s − 0.824·21-s − 1.64·22-s − 1.59·23-s − 0.364·24-s − 1.57·26-s − 1.37·27-s + 6.53·28-s + 2.22·29-s + 1.31·31-s + 4.39·32-s − 0.925·33-s − 0.291·34-s + ⋯ |
| L(s) = 1 | + 0.291·2-s + 0.133·3-s − 0.915·4-s + 0.0388·6-s − 1.34·7-s − 0.557·8-s − 0.982·9-s − 1.20·11-s − 0.122·12-s − 1.06·13-s − 0.392·14-s + 0.753·16-s − 0.172·17-s − 0.285·18-s − 1.43·19-s − 0.179·21-s − 0.351·22-s − 0.332·23-s − 0.0743·24-s − 0.309·26-s − 0.264·27-s + 1.23·28-s + 0.412·29-s + 0.235·31-s + 0.776·32-s − 0.161·33-s − 0.0500·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.004320807844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.004320807844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 197 | \( 1 + T \) |
| good | 2 | \( 1 - 0.411T + 2T^{2} \) |
| 3 | \( 1 - 0.231T + 3T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 4.00T + 11T^{2} \) |
| 13 | \( 1 + 3.83T + 13T^{2} \) |
| 17 | \( 1 + 0.709T + 17T^{2} \) |
| 19 | \( 1 + 6.25T + 19T^{2} \) |
| 23 | \( 1 + 1.59T + 23T^{2} \) |
| 29 | \( 1 - 2.22T + 29T^{2} \) |
| 31 | \( 1 - 1.31T + 31T^{2} \) |
| 37 | \( 1 + 5.52T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + 7.97T + 47T^{2} \) |
| 53 | \( 1 + 9.53T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 5.42T + 67T^{2} \) |
| 71 | \( 1 - 7.54T + 71T^{2} \) |
| 73 | \( 1 - 0.717T + 73T^{2} \) |
| 79 | \( 1 + 0.425T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−8.249082798980141219357933699998, −7.74824668387358754489691891587, −6.53008270739412503288411733592, −6.16580938178917925047873336210, −5.12666836931183465946116922213, −4.76250935045636058037798924688, −3.57638853613800261660574586756, −3.03629202119177778667743602958, −2.22317449680797359443373612053, −0.03131565452827996904895270360,
0.03131565452827996904895270360, 2.22317449680797359443373612053, 3.03629202119177778667743602958, 3.57638853613800261660574586756, 4.76250935045636058037798924688, 5.12666836931183465946116922213, 6.16580938178917925047873336210, 6.53008270739412503288411733592, 7.74824668387358754489691891587, 8.249082798980141219357933699998
