| L(s) = 1 | + 2.74·2-s + 3-s + 5.51·4-s + 0.315·5-s + 2.74·6-s + 9.64·8-s + 9-s + 0.866·10-s − 3.45·11-s + 5.51·12-s + 0.303·13-s + 0.315·15-s + 15.4·16-s − 5.96·17-s + 2.74·18-s + 8.54·19-s + 1.74·20-s − 9.47·22-s + 4.88·23-s + 9.64·24-s − 4.90·25-s + 0.831·26-s + 27-s + 10.1·29-s + 0.866·30-s − 6.98·31-s + 22.9·32-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 0.577·3-s + 2.75·4-s + 0.141·5-s + 1.11·6-s + 3.41·8-s + 0.333·9-s + 0.273·10-s − 1.04·11-s + 1.59·12-s + 0.0841·13-s + 0.0815·15-s + 3.85·16-s − 1.44·17-s + 0.646·18-s + 1.96·19-s + 0.389·20-s − 2.01·22-s + 1.01·23-s + 1.96·24-s − 0.980·25-s + 0.163·26-s + 0.192·27-s + 1.88·29-s + 0.158·30-s − 1.25·31-s + 4.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.742137188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.742137188\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 5 | \( 1 - 0.315T + 5T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 13 | \( 1 - 0.303T + 13T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 - 8.54T + 19T^{2} \) |
| 23 | \( 1 - 4.88T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 6.98T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 43 | \( 1 - 0.387T + 43T^{2} \) |
| 47 | \( 1 - 4.04T + 47T^{2} \) |
| 53 | \( 1 - 3.33T + 53T^{2} \) |
| 59 | \( 1 + 7.81T + 59T^{2} \) |
| 61 | \( 1 + 9.55T + 61T^{2} \) |
| 67 | \( 1 - 3.14T + 67T^{2} \) |
| 71 | \( 1 + 0.756T + 71T^{2} \) |
| 73 | \( 1 - 0.212T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 8.92T + 83T^{2} \) |
| 89 | \( 1 - 0.416T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.54400532156290061232081023989, −7.39839583044295120422813181184, −6.43424345815880249766168141849, −5.78521303733768358603121621408, −5.01508684812763160650882344766, −4.53970817195061802594095369858, −3.66665961196484400050526655846, −2.82525549090222440197453586630, −2.49147230481503767211463012558, −1.33130363526741436681324458158,
1.33130363526741436681324458158, 2.49147230481503767211463012558, 2.82525549090222440197453586630, 3.66665961196484400050526655846, 4.53970817195061802594095369858, 5.01508684812763160650882344766, 5.78521303733768358603121621408, 6.43424345815880249766168141849, 7.39839583044295120422813181184, 7.54400532156290061232081023989
