| L(s) = 1 | − 4.75·3-s + 22.0·5-s + 32.1·7-s − 4.38·9-s + 52.9·11-s − 23.5·13-s − 104.·15-s + 86.2·17-s − 95.1·19-s − 152.·21-s − 85.1·23-s + 361.·25-s + 149.·27-s + 29·29-s + 3.58·31-s − 251.·33-s + 708.·35-s + 368.·37-s + 111.·39-s − 257.·41-s + 196.·43-s − 96.7·45-s − 164.·47-s + 688.·49-s − 410.·51-s + 293.·53-s + 1.16e3·55-s + ⋯ |
| L(s) = 1 | − 0.915·3-s + 1.97·5-s + 1.73·7-s − 0.162·9-s + 1.45·11-s − 0.501·13-s − 1.80·15-s + 1.23·17-s − 1.14·19-s − 1.58·21-s − 0.771·23-s + 2.89·25-s + 1.06·27-s + 0.185·29-s + 0.0207·31-s − 1.32·33-s + 3.42·35-s + 1.63·37-s + 0.458·39-s − 0.981·41-s + 0.698·43-s − 0.320·45-s − 0.511·47-s + 2.00·49-s − 1.12·51-s + 0.760·53-s + 2.86·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.458717134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.458717134\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 4.75T + 27T^{2} \) |
| 5 | \( 1 - 22.0T + 125T^{2} \) |
| 7 | \( 1 - 32.1T + 343T^{2} \) |
| 11 | \( 1 - 52.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 86.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 95.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 85.1T + 1.21e4T^{2} \) |
| 31 | \( 1 - 3.58T + 2.97e4T^{2} \) |
| 37 | \( 1 - 368.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 257.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 196.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 164.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 293.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 73.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 666.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 155.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 400.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 300.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 510.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 637.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 129.T + 9.12e5T^{2} \) |
| 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
−8.890591258569949497562846530731, −8.274978908772284812647016142513, −7.06872997598060346242943911237, −6.18787022423171160291188311796, −5.74138362036240992091418218312, −5.03973071389066130243182080583, −4.26193599715034746692441197859, −2.55220833135160556261865683247, −1.66284608745555839035677448315, −1.02270431722260773031722861875,
1.02270431722260773031722861875, 1.66284608745555839035677448315, 2.55220833135160556261865683247, 4.26193599715034746692441197859, 5.03973071389066130243182080583, 5.74138362036240992091418218312, 6.18787022423171160291188311796, 7.06872997598060346242943911237, 8.274978908772284812647016142513, 8.890591258569949497562846530731
