| L(s) = 1 | + 64.0·5-s + 65.8·7-s − 635.·11-s + 467.·13-s − 522.·17-s − 361·19-s + 3.22e3·23-s + 981.·25-s − 6.97e3·29-s − 3.38e3·31-s + 4.21e3·35-s − 1.34e4·37-s − 7.10e3·41-s − 1.40e4·43-s + 1.44e3·47-s − 1.24e4·49-s + 3.71e4·53-s − 4.07e4·55-s − 3.78e4·59-s + 3.96e4·61-s + 2.99e4·65-s − 1.10e4·67-s − 9.09e3·71-s + 2.99e4·73-s − 4.18e4·77-s − 3.32e4·79-s + 5.07e4·83-s + ⋯ |
| L(s) = 1 | + 1.14·5-s + 0.507·7-s − 1.58·11-s + 0.767·13-s − 0.438·17-s − 0.229·19-s + 1.26·23-s + 0.313·25-s − 1.54·29-s − 0.633·31-s + 0.581·35-s − 1.61·37-s − 0.660·41-s − 1.15·43-s + 0.0953·47-s − 0.742·49-s + 1.81·53-s − 1.81·55-s − 1.41·59-s + 1.36·61-s + 0.880·65-s − 0.299·67-s − 0.214·71-s + 0.657·73-s − 0.804·77-s − 0.599·79-s + 0.808·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 361T \) |
| good | 5 | \( 1 - 64.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 65.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 635.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 467.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 522.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.34e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.10e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.40e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.44e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.71e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.96e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.09e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.28e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.11e5T + 8.58e9T^{2} \) |
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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
−9.231431314642824999989108285309, −8.531665516995655672922298839471, −7.54278292809613097764758198804, −6.56903321195575120154886164823, −5.45385117200077294756647350060, −5.08078255518361660365427284538, −3.54546239632540110426404819866, −2.33946653448576190171848282646, −1.54083994032539700368871739180, 0,
1.54083994032539700368871739180, 2.33946653448576190171848282646, 3.54546239632540110426404819866, 5.08078255518361660365427284538, 5.45385117200077294756647350060, 6.56903321195575120154886164823, 7.54278292809613097764758198804, 8.531665516995655672922298839471, 9.231431314642824999989108285309
