| L(s) = 1 | − 1.26·2-s − 6.39·4-s − 3.92·5-s + 18.2·8-s + 4.98·10-s − 51.7·11-s − 67.5·13-s + 27.9·16-s + 63.4·17-s + 71.9·19-s + 25.1·20-s + 65.5·22-s − 147.·23-s − 109.·25-s + 85.6·26-s − 117.·29-s − 54.7·31-s − 181.·32-s − 80.5·34-s + 9.70·37-s − 91.1·38-s − 71.6·40-s − 236.·41-s − 489.·43-s + 330.·44-s + 187.·46-s + 613.·47-s + ⋯ |
| L(s) = 1 | − 0.448·2-s − 0.799·4-s − 0.351·5-s + 0.806·8-s + 0.157·10-s − 1.41·11-s − 1.44·13-s + 0.437·16-s + 0.905·17-s + 0.868·19-s + 0.280·20-s + 0.635·22-s − 1.33·23-s − 0.876·25-s + 0.646·26-s − 0.753·29-s − 0.316·31-s − 1.00·32-s − 0.406·34-s + 0.0431·37-s − 0.389·38-s − 0.283·40-s − 0.900·41-s − 1.73·43-s + 1.13·44-s + 0.600·46-s + 1.90·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3535147166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3535147166\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.26T + 8T^{2} \) |
| 5 | \( 1 + 3.92T + 125T^{2} \) |
| 11 | \( 1 + 51.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 67.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 117.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 54.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 9.70T + 5.06e4T^{2} \) |
| 41 | \( 1 + 236.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 489.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 613.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 316.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 2.43T + 2.05e5T^{2} \) |
| 61 | \( 1 + 482.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 646.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 459.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 137.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 816.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 255.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 62.9T + 9.12e5T^{2} \) |
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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
−9.454797620452498735747298468023, −8.255240893443178631128607904750, −7.78191408827438268485925146769, −7.23983179163634404191904145114, −5.61261705391939498576129662142, −5.15649706773807242275127136286, −4.14894315093212786988719565516, −3.11144669706821286879265207432, −1.85913980337783281874636711339, −0.31136106903687004700543830908,
0.31136106903687004700543830908, 1.85913980337783281874636711339, 3.11144669706821286879265207432, 4.14894315093212786988719565516, 5.15649706773807242275127136286, 5.61261705391939498576129662142, 7.23983179163634404191904145114, 7.78191408827438268485925146769, 8.255240893443178631128607904750, 9.454797620452498735747298468023
