| L(s) = 1 | − 2-s − 3.00·3-s + 4-s − 5-s + 3.00·6-s − 1.41·7-s − 8-s + 6.03·9-s + 10-s + 11-s − 3.00·12-s − 2.39·13-s + 1.41·14-s + 3.00·15-s + 16-s − 7.53·17-s − 6.03·18-s − 1.50·19-s − 20-s + 4.25·21-s − 22-s + 4.87·23-s + 3.00·24-s + 25-s + 2.39·26-s − 9.10·27-s − 1.41·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 0.5·4-s − 0.447·5-s + 1.22·6-s − 0.534·7-s − 0.353·8-s + 2.01·9-s + 0.316·10-s + 0.301·11-s − 0.867·12-s − 0.663·13-s + 0.378·14-s + 0.775·15-s + 0.250·16-s − 1.82·17-s − 1.42·18-s − 0.344·19-s − 0.223·20-s + 0.927·21-s − 0.213·22-s + 1.01·23-s + 0.613·24-s + 0.200·25-s + 0.468·26-s − 1.75·27-s − 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1790172633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1790172633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 3.00T + 3T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 13 | \( 1 + 2.39T + 13T^{2} \) |
| 17 | \( 1 + 7.53T + 17T^{2} \) |
| 19 | \( 1 + 1.50T + 19T^{2} \) |
| 23 | \( 1 - 4.87T + 23T^{2} \) |
| 29 | \( 1 - 3.11T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 1.00T + 37T^{2} \) |
| 41 | \( 1 - 5.33T + 41T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 5.53T + 67T^{2} \) |
| 71 | \( 1 - 2.08T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.299377874607346790384059134643, −7.21877244989141443557408155063, −6.88048838984718068465850246910, −6.31149776059416715475864064603, −5.47984885505957162417311225775, −4.69091262523510274306710665377, −4.00598834478676088842616345481, −2.71131073643666434359794331623, −1.51558821530680816760858499382, −0.28988349062465364751670821501,
0.28988349062465364751670821501, 1.51558821530680816760858499382, 2.71131073643666434359794331623, 4.00598834478676088842616345481, 4.69091262523510274306710665377, 5.47984885505957162417311225775, 6.31149776059416715475864064603, 6.88048838984718068465850246910, 7.21877244989141443557408155063, 8.299377874607346790384059134643
