| L(s) = 1 | − 4.59·2-s + 3·3-s + 13.1·4-s − 13.7·6-s − 20.6·7-s − 23.4·8-s + 9·9-s + 11·11-s + 39.3·12-s + 15.6·13-s + 94.8·14-s + 3.04·16-s − 72.9·17-s − 41.3·18-s + 61.0·19-s − 61.9·21-s − 50.5·22-s + 13.6·23-s − 70.4·24-s − 71.9·26-s + 27·27-s − 270.·28-s − 31.4·29-s − 243.·31-s + 173.·32-s + 33·33-s + 335.·34-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 0.577·3-s + 1.63·4-s − 0.937·6-s − 1.11·7-s − 1.03·8-s + 0.333·9-s + 0.301·11-s + 0.946·12-s + 0.334·13-s + 1.81·14-s + 0.0475·16-s − 1.04·17-s − 0.541·18-s + 0.737·19-s − 0.643·21-s − 0.489·22-s + 0.123·23-s − 0.599·24-s − 0.542·26-s + 0.192·27-s − 1.82·28-s − 0.201·29-s − 1.40·31-s + 0.961·32-s + 0.174·33-s + 1.69·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8116707119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8116707119\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 4.59T + 8T^{2} \) |
| 7 | \( 1 + 20.6T + 343T^{2} \) |
| 13 | \( 1 - 15.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 31.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 65.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 109.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 121.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 519.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 542.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 109.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 89.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 488.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 837.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 351.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 831.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 426.T + 9.12e5T^{2} \) |
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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
−9.521029819323548047546794083204, −9.113811668177472805863150717984, −8.430494558339319438543963743224, −7.33218434657183980995933289164, −6.88208156038497257773965352679, −5.80740772733824205105383991124, −4.11816195311908895171349074546, −2.98899358661704084035780954180, −1.90607871882990355994974059123, −0.61226534813122542488314598292,
0.61226534813122542488314598292, 1.90607871882990355994974059123, 2.98899358661704084035780954180, 4.11816195311908895171349074546, 5.80740772733824205105383991124, 6.88208156038497257773965352679, 7.33218434657183980995933289164, 8.430494558339319438543963743224, 9.113811668177472805863150717984, 9.521029819323548047546794083204
