| L(s) = 1 | − 1.70·2-s − 3·3-s − 5.10·4-s + 5·5-s + 5.10·6-s + 22.2·8-s + 9·9-s − 8.50·10-s + 37.4·11-s + 15.3·12-s − 29.0·13-s − 15·15-s + 2.89·16-s − 58.4·17-s − 15.3·18-s + 54.5·19-s − 25.5·20-s − 63.6·22-s + 161.·23-s − 66.8·24-s + 25·25-s + 49.3·26-s − 27·27-s + 137.·29-s + 25.5·30-s − 154.·31-s − 183.·32-s + ⋯ |
| L(s) = 1 | − 0.601·2-s − 0.577·3-s − 0.638·4-s + 0.447·5-s + 0.347·6-s + 0.985·8-s + 0.333·9-s − 0.269·10-s + 1.02·11-s + 0.368·12-s − 0.619·13-s − 0.258·15-s + 0.0452·16-s − 0.833·17-s − 0.200·18-s + 0.659·19-s − 0.285·20-s − 0.616·22-s + 1.46·23-s − 0.568·24-s + 0.200·25-s + 0.372·26-s − 0.192·27-s + 0.880·29-s + 0.155·30-s − 0.896·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.009278437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.009278437\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.70T + 8T^{2} \) |
| 11 | \( 1 - 37.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 29.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 58.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 350.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 518.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 542.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 305.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 14.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 171.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 551.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 120.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 941.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 377.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 677.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−9.958879774976066977732556786045, −9.091875961957082198469817865672, −8.619662345738057846624240375741, −7.23424147500228175669407005459, −6.69285609841721164681306558959, −5.33976194384292118421124142230, −4.74045759646112783460902802576, −3.52199884634867288140851344149, −1.79147931048963323611244451144, −0.67434224067588292521990695935,
0.67434224067588292521990695935, 1.79147931048963323611244451144, 3.52199884634867288140851344149, 4.74045759646112783460902802576, 5.33976194384292118421124142230, 6.69285609841721164681306558959, 7.23424147500228175669407005459, 8.619662345738057846624240375741, 9.091875961957082198469817865672, 9.958879774976066977732556786045
