| L(s) = 1 | + 0.381·2-s − 1.85·4-s − 1.23·5-s + 7-s − 1.47·8-s − 0.472·10-s + 6.47·13-s + 0.381·14-s + 3.14·16-s − 4.85·17-s − 19-s + 2.29·20-s + 4.61·23-s − 3.47·25-s + 2.47·26-s − 1.85·28-s + 4.85·29-s + 0.618·31-s + 4.14·32-s − 1.85·34-s − 1.23·35-s − 5.09·37-s − 0.381·38-s + 1.81·40-s + 2.52·41-s − 1.85·43-s + 1.76·46-s + ⋯ |
| L(s) = 1 | + 0.270·2-s − 0.927·4-s − 0.552·5-s + 0.377·7-s − 0.520·8-s − 0.149·10-s + 1.79·13-s + 0.102·14-s + 0.786·16-s − 1.17·17-s − 0.229·19-s + 0.512·20-s + 0.962·23-s − 0.694·25-s + 0.484·26-s − 0.350·28-s + 0.901·29-s + 0.111·31-s + 0.732·32-s − 0.317·34-s − 0.208·35-s − 0.836·37-s − 0.0619·38-s + 0.287·40-s + 0.394·41-s − 0.282·43-s + 0.260·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.588142356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.588142356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 - 0.618T + 31T^{2} \) |
| 37 | \( 1 + 5.09T + 37T^{2} \) |
| 41 | \( 1 - 2.52T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 - 1.61T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 + 2.90T + 71T^{2} \) |
| 73 | \( 1 - 0.145T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 9.76T + 83T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
| 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
−7.83930843392623887224758119002, −6.92403122108201704482542316116, −6.23355808594178038791179289833, −5.57669854346149031837300671958, −4.75748317279031782663460580074, −4.17675303165656506428508129892, −3.66283417286835261222166689664, −2.81333108000117071862557887263, −1.56784041589154444039259403380, −0.60855443679800552377300366260,
0.60855443679800552377300366260, 1.56784041589154444039259403380, 2.81333108000117071862557887263, 3.66283417286835261222166689664, 4.17675303165656506428508129892, 4.75748317279031782663460580074, 5.57669854346149031837300671958, 6.23355808594178038791179289833, 6.92403122108201704482542316116, 7.83930843392623887224758119002
