| L(s) = 1 | − 4.45·2-s + 11.8·4-s + 14.0·5-s + 23.7·7-s − 16.9·8-s − 62.4·10-s + 36.5·11-s + 56.5·13-s − 105.·14-s − 18.9·16-s − 7.01·17-s − 45.1·19-s + 165.·20-s − 162.·22-s + 174.·23-s + 71.8·25-s − 251.·26-s + 280.·28-s − 226.·29-s + 97.8·31-s + 220.·32-s + 31.2·34-s + 332.·35-s − 338.·37-s + 200.·38-s − 238.·40-s − 481.·41-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 1.47·4-s + 1.25·5-s + 1.27·7-s − 0.750·8-s − 1.97·10-s + 1.00·11-s + 1.20·13-s − 2.01·14-s − 0.295·16-s − 0.100·17-s − 0.545·19-s + 1.85·20-s − 1.57·22-s + 1.57·23-s + 0.574·25-s − 1.89·26-s + 1.89·28-s − 1.44·29-s + 0.567·31-s + 1.21·32-s + 0.157·34-s + 1.60·35-s − 1.50·37-s + 0.857·38-s − 0.941·40-s − 1.83·41-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
| \(L(2)\) |
\(\approx\) |
\(1.955270643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.955270643\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.45T + 8T^{2} \) |
| 5 | \( 1 - 14.0T + 125T^{2} \) |
| 7 | \( 1 - 23.7T + 343T^{2} \) |
| 11 | \( 1 - 36.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 56.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.01T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 174.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 97.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 338.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 481.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 58.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 407.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 302.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 785.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 81.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 813.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 437.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 631.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 757.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 650.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 806.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.21e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.644940591009484742900573353843, −8.464118897617234385753456624688, −7.21208842488809274289692983088, −6.69210146306010427847444859676, −5.75272090361548140043531997148, −4.90142960489505012784539834751, −3.66940039320387172229740638975, −2.13039468649935503528056666967, −1.61073975325724271178502758208, −0.890925488961552131411136107880,
0.890925488961552131411136107880, 1.61073975325724271178502758208, 2.13039468649935503528056666967, 3.66940039320387172229740638975, 4.90142960489505012784539834751, 5.75272090361548140043531997148, 6.69210146306010427847444859676, 7.21208842488809274289692983088, 8.464118897617234385753456624688, 8.644940591009484742900573353843
