L(s) = 1 | − 2-s − 3-s + 4-s + 1.73·5-s + 6-s + 1.26·7-s − 8-s + 9-s − 1.73·10-s + 1.26·11-s − 12-s − 1.26·14-s − 1.73·15-s + 16-s + 5.19·17-s − 18-s + 4.73·19-s + 1.73·20-s − 1.26·21-s − 1.26·22-s − 8.19·23-s + 24-s − 2.00·25-s − 27-s + 1.26·28-s − 3·29-s + 1.73·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.774·5-s + 0.408·6-s + 0.479·7-s − 0.353·8-s + 0.333·9-s − 0.547·10-s + 0.382·11-s − 0.288·12-s − 0.338·14-s − 0.447·15-s + 0.250·16-s + 1.26·17-s − 0.235·18-s + 1.08·19-s + 0.387·20-s − 0.276·21-s − 0.270·22-s − 1.70·23-s + 0.204·24-s − 0.400·25-s − 0.192·27-s + 0.239·28-s − 0.557·29-s + 0.316·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.230447843\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.230447843\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 + 8.19T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 6.46T + 41T^{2} \) |
| 43 | \( 1 + 4.19T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 7.26T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 - 5.66T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−9.808033411500677063025716791414, −9.505359062198013910657283906976, −8.130669724493807723440549167198, −7.67577756352858124743759631169, −6.44303574898647444613346554543, −5.85344792625975773335314123079, −4.95380999155144329793844369258, −3.59922580593167702004495469519, −2.13545536210567694241762561136, −1.05273906250767299757696084873,
1.05273906250767299757696084873, 2.13545536210567694241762561136, 3.59922580593167702004495469519, 4.95380999155144329793844369258, 5.85344792625975773335314123079, 6.44303574898647444613346554543, 7.67577756352858124743759631169, 8.130669724493807723440549167198, 9.505359062198013910657283906976, 9.808033411500677063025716791414