| L(s) = 1 | + 36.7·2-s + 155.·3-s + 837.·4-s + 1.72e3·5-s + 5.70e3·6-s − 2.40e3·7-s + 1.19e4·8-s + 4.46e3·9-s + 6.35e4·10-s + 6.07e3·11-s + 1.30e5·12-s + 2.85e4·13-s − 8.81e4·14-s + 2.68e5·15-s + 1.02e4·16-s + 5.56e5·17-s + 1.64e5·18-s + 5.05e5·19-s + 1.44e6·20-s − 3.73e5·21-s + 2.23e5·22-s + 3.77e5·23-s + 1.85e6·24-s + 1.03e6·25-s + 1.04e6·26-s − 2.36e6·27-s − 2.01e6·28-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.10·3-s + 1.63·4-s + 1.23·5-s + 1.79·6-s − 0.377·7-s + 1.03·8-s + 0.227·9-s + 2.00·10-s + 0.125·11-s + 1.81·12-s + 0.277·13-s − 0.613·14-s + 1.37·15-s + 0.0392·16-s + 1.61·17-s + 0.368·18-s + 0.889·19-s + 2.02·20-s − 0.418·21-s + 0.203·22-s + 0.281·23-s + 1.14·24-s + 0.531·25-s + 0.450·26-s − 0.856·27-s − 0.618·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(9.308130277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.308130277\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 2.40e3T \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 - 36.7T + 512T^{2} \) |
| 3 | \( 1 - 155.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.72e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 6.07e3T + 2.35e9T^{2} \) |
| 17 | \( 1 - 5.56e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.05e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.77e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.91e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.44e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.79e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.69e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.34e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.47e5T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.00e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.64e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.36e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.65e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.36e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.55e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.25e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.20e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.01e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.27e8T + 7.60e17T^{2} \) |
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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
−12.80181687735801603821243186465, −11.57438579805474733345335092633, −10.00007296983652938152927804580, −9.089108324783431500974362621209, −7.49640290599776296197176330403, −6.07226359038446367226842385057, −5.30932359268379778061640772868, −3.61131908578220358616921331004, −2.87343959318121645409680473878, −1.67282364294419358550121029240,
1.67282364294419358550121029240, 2.87343959318121645409680473878, 3.61131908578220358616921331004, 5.30932359268379778061640772868, 6.07226359038446367226842385057, 7.49640290599776296197176330403, 9.089108324783431500974362621209, 10.00007296983652938152927804580, 11.57438579805474733345335092633, 12.80181687735801603821243186465
