| L(s) = 1 | − 3.54·2-s + 3·3-s + 4.59·4-s − 5·5-s − 10.6·6-s − 21.2·7-s + 12.0·8-s + 9·9-s + 17.7·10-s − 40.9·11-s + 13.7·12-s − 8.89·13-s + 75.4·14-s − 15·15-s − 79.6·16-s − 7.68·17-s − 31.9·18-s + 40.7·19-s − 22.9·20-s − 63.8·21-s + 145.·22-s − 80.7·23-s + 36.2·24-s + 25·25-s + 31.5·26-s + 27·27-s − 97.6·28-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.577·3-s + 0.574·4-s − 0.447·5-s − 0.724·6-s − 1.14·7-s + 0.534·8-s + 0.333·9-s + 0.561·10-s − 1.12·11-s + 0.331·12-s − 0.189·13-s + 1.44·14-s − 0.258·15-s − 1.24·16-s − 0.109·17-s − 0.418·18-s + 0.492·19-s − 0.256·20-s − 0.662·21-s + 1.40·22-s − 0.731·23-s + 0.308·24-s + 0.200·25-s + 0.238·26-s + 0.192·27-s − 0.659·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6070948371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6070948371\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 3.54T + 8T^{2} \) |
| 7 | \( 1 + 21.2T + 343T^{2} \) |
| 11 | \( 1 + 40.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.89T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.68T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 80.7T + 1.21e4T^{2} \) |
| 31 | \( 1 - 82.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 53.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 17.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 23.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 399.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 388.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 399.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 92.2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 979.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 62.2T + 4.93e5T^{2} \) |
| 83 | \( 1 - 163.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.26e3T + 9.12e5T^{2} \) |
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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
−10.24030449089915120947919333313, −9.835479291848730766500807362793, −8.935761487408207520141172244093, −8.047130212591606207758372479493, −7.46055508867752003584483980004, −6.39473236144267207853432714119, −4.83765353122445964875511505489, −3.48470079638548784527897010141, −2.31670120379676754200170634414, −0.56358524183387357071569063209,
0.56358524183387357071569063209, 2.31670120379676754200170634414, 3.48470079638548784527897010141, 4.83765353122445964875511505489, 6.39473236144267207853432714119, 7.46055508867752003584483980004, 8.047130212591606207758372479493, 8.935761487408207520141172244093, 9.835479291848730766500807362793, 10.24030449089915120947919333313
