| L(s) = 1 | − 4.70·2-s + 14.1·4-s − 5·5-s − 28.7·8-s + 23.5·10-s − 24.5·11-s + 35.0·13-s + 22.1·16-s − 18.4·17-s + 67.4·19-s − 70.5·20-s + 115.·22-s + 145.·23-s + 25·25-s − 164.·26-s − 214.·29-s + 88.6·31-s + 125.·32-s + 86.5·34-s + 162.·37-s − 316.·38-s + 143.·40-s − 337.·41-s + 122.·43-s − 346.·44-s − 684.·46-s + 354.·47-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.76·4-s − 0.447·5-s − 1.26·8-s + 0.743·10-s − 0.674·11-s + 0.747·13-s + 0.345·16-s − 0.262·17-s + 0.813·19-s − 0.788·20-s + 1.12·22-s + 1.32·23-s + 0.200·25-s − 1.24·26-s − 1.37·29-s + 0.513·31-s + 0.694·32-s + 0.436·34-s + 0.720·37-s − 1.35·38-s + 0.567·40-s − 1.28·41-s + 0.433·43-s − 1.18·44-s − 2.19·46-s + 1.09·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7203154080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7203154080\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.70T + 8T^{2} \) |
| 11 | \( 1 + 24.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 35.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 501.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 708.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 907.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 430.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 41.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 890.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.47e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 555.T + 9.12e5T^{2} \) |
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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
−8.765023127653518868449475180324, −8.011307591654896696318463065031, −7.43039046950796777063564665707, −6.77500121526243522454268513281, −5.76951821820016139012997102054, −4.75093120025812212571984131523, −3.50589041345024107644173498813, −2.55841064261338600173888301568, −1.41669654627431476667269737316, −0.52963839797465127935562734935,
0.52963839797465127935562734935, 1.41669654627431476667269737316, 2.55841064261338600173888301568, 3.50589041345024107644173498813, 4.75093120025812212571984131523, 5.76951821820016139012997102054, 6.77500121526243522454268513281, 7.43039046950796777063564665707, 8.011307591654896696318463065031, 8.765023127653518868449475180324
