| L(s) = 1 | + 2-s + 3.04·3-s + 4-s − 2.77·5-s + 3.04·6-s + 1.86·7-s + 8-s + 6.29·9-s − 2.77·10-s + 0.985·11-s + 3.04·12-s + 6.02·13-s + 1.86·14-s − 8.45·15-s + 16-s + 1.84·17-s + 6.29·18-s + 19-s − 2.77·20-s + 5.68·21-s + 0.985·22-s − 3.53·23-s + 3.04·24-s + 2.69·25-s + 6.02·26-s + 10.0·27-s + 1.86·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.75·3-s + 0.5·4-s − 1.24·5-s + 1.24·6-s + 0.704·7-s + 0.353·8-s + 2.09·9-s − 0.877·10-s + 0.297·11-s + 0.879·12-s + 1.67·13-s + 0.498·14-s − 2.18·15-s + 0.250·16-s + 0.447·17-s + 1.48·18-s + 0.229·19-s − 0.620·20-s + 1.24·21-s + 0.210·22-s − 0.738·23-s + 0.622·24-s + 0.538·25-s + 1.18·26-s + 1.93·27-s + 0.352·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.635386196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.635386196\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 5 | \( 1 + 2.77T + 5T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 - 0.985T + 11T^{2} \) |
| 13 | \( 1 - 6.02T + 13T^{2} \) |
| 17 | \( 1 - 1.84T + 17T^{2} \) |
| 23 | \( 1 + 3.53T + 23T^{2} \) |
| 29 | \( 1 + 2.47T + 29T^{2} \) |
| 31 | \( 1 - 1.02T + 31T^{2} \) |
| 37 | \( 1 - 9.69T + 37T^{2} \) |
| 41 | \( 1 + 5.76T + 41T^{2} \) |
| 43 | \( 1 - 5.13T + 43T^{2} \) |
| 47 | \( 1 - 5.54T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 7.81T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 6.84T + 67T^{2} \) |
| 71 | \( 1 - 8.67T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 1.57T + 79T^{2} \) |
| 83 | \( 1 + 0.321T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 - 9.39T + 97T^{2} \) |
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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
−7.78401063609348040317464062863, −7.55636938108808918004328303213, −6.53567549676618573095617233978, −5.75758828008982452595816825933, −4.51839737285328835932530814656, −4.15575098801853482984800628829, −3.49738274565997169477337910277, −3.04084261885726407165088040258, −1.93899284294211063878942070535, −1.16982029419837496995172759788,
1.16982029419837496995172759788, 1.93899284294211063878942070535, 3.04084261885726407165088040258, 3.49738274565997169477337910277, 4.15575098801853482984800628829, 4.51839737285328835932530814656, 5.75758828008982452595816825933, 6.53567549676618573095617233978, 7.55636938108808918004328303213, 7.78401063609348040317464062863
