| L(s) = 1 | − 2·2-s − 1.68·3-s + 4·4-s + 3.37·6-s − 8·8-s − 24.1·9-s + 30.0·11-s − 6.74·12-s + 17.7·13-s + 16·16-s − 32.7·17-s + 48.3·18-s − 57.2·19-s − 60.1·22-s + 54.0·23-s + 13.4·24-s − 35.5·26-s + 86.2·27-s − 164.·29-s + 230.·31-s − 32·32-s − 50.7·33-s + 65.4·34-s − 96.6·36-s − 4.00·37-s + 114.·38-s − 29.9·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.324·3-s + 0.5·4-s + 0.229·6-s − 0.353·8-s − 0.894·9-s + 0.824·11-s − 0.162·12-s + 0.379·13-s + 0.250·16-s − 0.467·17-s + 0.632·18-s − 0.691·19-s − 0.583·22-s + 0.490·23-s + 0.114·24-s − 0.268·26-s + 0.614·27-s − 1.05·29-s + 1.33·31-s − 0.176·32-s − 0.267·33-s + 0.330·34-s − 0.447·36-s − 0.0178·37-s + 0.488·38-s − 0.123·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.68T + 27T^{2} \) |
| 11 | \( 1 - 30.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 57.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 54.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 230.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 4.00T + 5.06e4T^{2} \) |
| 41 | \( 1 + 120.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 91.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 62.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 39.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 661.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 254.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 163.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 38.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 253.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.09e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 500.T + 9.12e5T^{2} \) |
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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
−8.396099526513566500417625443429, −7.52294405652290639210833523119, −6.54052132637487506018343328423, −6.16891241515755178188659937697, −5.17922200029100571505072453514, −4.14734081710576141289791406576, −3.14337141595646777281789859458, −2.14271602132647618290197168914, −1.03877343463403589510217597878, 0,
1.03877343463403589510217597878, 2.14271602132647618290197168914, 3.14337141595646777281789859458, 4.14734081710576141289791406576, 5.17922200029100571505072453514, 6.16891241515755178188659937697, 6.54052132637487506018343328423, 7.52294405652290639210833523119, 8.396099526513566500417625443429
