| L(s) = 1 | − 2.23·2-s − 3.01·4-s − 1.56·5-s − 14.0·7-s + 24.5·8-s + 3.48·10-s − 21.7·11-s + 37.8·13-s + 31.3·14-s − 30.7·16-s + 37.1·17-s + 63.4·19-s + 4.70·20-s + 48.6·22-s + 98.7·23-s − 122.·25-s − 84.4·26-s + 42.3·28-s + 246.·29-s + 132.·31-s − 128.·32-s − 82.9·34-s + 21.9·35-s − 387.·37-s − 141.·38-s − 38.3·40-s + 256.·41-s + ⋯ |
| L(s) = 1 | − 0.789·2-s − 0.376·4-s − 0.139·5-s − 0.758·7-s + 1.08·8-s + 0.110·10-s − 0.596·11-s + 0.807·13-s + 0.598·14-s − 0.481·16-s + 0.530·17-s + 0.766·19-s + 0.0526·20-s + 0.471·22-s + 0.895·23-s − 0.980·25-s − 0.637·26-s + 0.285·28-s + 1.57·29-s + 0.768·31-s − 0.707·32-s − 0.418·34-s + 0.105·35-s − 1.72·37-s − 0.604·38-s − 0.151·40-s + 0.978·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9060891031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9060891031\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 2.23T + 8T^{2} \) |
| 5 | \( 1 + 1.56T + 125T^{2} \) |
| 7 | \( 1 + 14.0T + 343T^{2} \) |
| 11 | \( 1 + 21.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 63.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 98.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 246.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 387.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 256.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 419.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 100.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 37.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 331.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 440.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 372.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 123.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 776.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 981.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 32.5T + 9.12e5T^{2} \) |
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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
−8.653407972228198708674879487768, −8.172367595193781548788550508249, −7.33615280271261654671209634529, −6.55379186563733444713783343485, −5.53916163975186319116341130532, −4.75567674953292545540441431603, −3.69381660716851568888415184169, −2.90803926171614314682801453609, −1.44825497101831436550863794418, −0.52843585314088392591092940909,
0.52843585314088392591092940909, 1.44825497101831436550863794418, 2.90803926171614314682801453609, 3.69381660716851568888415184169, 4.75567674953292545540441431603, 5.53916163975186319116341130532, 6.55379186563733444713783343485, 7.33615280271261654671209634529, 8.172367595193781548788550508249, 8.653407972228198708674879487768
