| L(s) = 1 | − 2-s + 2.50·3-s + 4-s + 3.78·5-s − 2.50·6-s − 1.89·7-s − 8-s + 3.29·9-s − 3.78·10-s + 6.00·11-s + 2.50·12-s − 0.142·13-s + 1.89·14-s + 9.50·15-s + 16-s − 1.74·17-s − 3.29·18-s + 7.93·19-s + 3.78·20-s − 4.74·21-s − 6.00·22-s − 23-s − 2.50·24-s + 9.36·25-s + 0.142·26-s + 0.727·27-s − 1.89·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.44·3-s + 0.5·4-s + 1.69·5-s − 1.02·6-s − 0.714·7-s − 0.353·8-s + 1.09·9-s − 1.19·10-s + 1.81·11-s + 0.723·12-s − 0.0394·13-s + 0.505·14-s + 2.45·15-s + 0.250·16-s − 0.424·17-s − 0.775·18-s + 1.82·19-s + 0.847·20-s − 1.03·21-s − 1.28·22-s − 0.208·23-s − 0.511·24-s + 1.87·25-s + 0.0278·26-s + 0.139·27-s − 0.357·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.837494945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.837494945\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 - 3.78T + 5T^{2} \) |
| 7 | \( 1 + 1.89T + 7T^{2} \) |
| 11 | \( 1 - 6.00T + 11T^{2} \) |
| 13 | \( 1 + 0.142T + 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 - 7.93T + 19T^{2} \) |
| 29 | \( 1 - 2.03T + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 0.870T + 43T^{2} \) |
| 47 | \( 1 + 3.10T + 47T^{2} \) |
| 53 | \( 1 - 2.61T + 53T^{2} \) |
| 59 | \( 1 - 0.307T + 59T^{2} \) |
| 61 | \( 1 - 5.55T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 8.62T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 5.26T + 89T^{2} \) |
| 97 | \( 1 + 0.273T + 97T^{2} \) |
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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
−8.405030101669758589216710868517, −7.29015710946413172952544051218, −6.87502793490709258420233249265, −6.11433470251892326673331391913, −5.47210698507358522881939131986, −4.16349668899612546779012896554, −3.27620451378108852105037930009, −2.70620068721114735543225632231, −1.79192048483558984637392200800, −1.20213031830694803473038629505,
1.20213031830694803473038629505, 1.79192048483558984637392200800, 2.70620068721114735543225632231, 3.27620451378108852105037930009, 4.16349668899612546779012896554, 5.47210698507358522881939131986, 6.11433470251892326673331391913, 6.87502793490709258420233249265, 7.29015710946413172952544051218, 8.405030101669758589216710868517
