| L(s) = 1 | + 2-s + 1.61·3-s + 4-s + 1.23·5-s + 1.61·6-s + 3.23·7-s + 8-s − 0.381·9-s + 1.23·10-s + 11-s + 1.61·12-s − 0.854·13-s + 3.23·14-s + 2.00·15-s + 16-s + 0.763·17-s − 0.381·18-s + 1.85·19-s + 1.23·20-s + 5.23·21-s + 22-s + 2·23-s + 1.61·24-s − 3.47·25-s − 0.854·26-s − 5.47·27-s + 3.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.934·3-s + 0.5·4-s + 0.552·5-s + 0.660·6-s + 1.22·7-s + 0.353·8-s − 0.127·9-s + 0.390·10-s + 0.301·11-s + 0.467·12-s − 0.236·13-s + 0.864·14-s + 0.516·15-s + 0.250·16-s + 0.185·17-s − 0.0900·18-s + 0.425·19-s + 0.276·20-s + 1.14·21-s + 0.213·22-s + 0.417·23-s + 0.330·24-s − 0.694·25-s − 0.167·26-s − 1.05·27-s + 0.611·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1474 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1474 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.310856098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.310856098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 13 | \( 1 + 0.854T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 + 3.38T + 31T^{2} \) |
| 37 | \( 1 - 1.14T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 4.76T + 43T^{2} \) |
| 47 | \( 1 - 8.76T + 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−9.308018786529995840725112393482, −8.776958444780189431030496196100, −7.70072067029257885292313364068, −7.36585319598698788417869973772, −5.92870284229866561399675564250, −5.40246969615894539996111836433, −4.35602153984041629118806922037, −3.46178655200750915808725869609, −2.38349148889661955832382093658, −1.61406431983922960611906333089,
1.61406431983922960611906333089, 2.38349148889661955832382093658, 3.46178655200750915808725869609, 4.35602153984041629118806922037, 5.40246969615894539996111836433, 5.92870284229866561399675564250, 7.36585319598698788417869973772, 7.70072067029257885292313364068, 8.776958444780189431030496196100, 9.308018786529995840725112393482
