| L(s) = 1 | + 69.8·3-s − 170.·5-s + 1.20e4·7-s − 1.48e4·9-s + 3.62e4·11-s − 2.85e4·13-s − 1.19e4·15-s − 5.23e5·17-s − 7.06e5·19-s + 8.39e5·21-s + 2.66e4·23-s − 1.92e6·25-s − 2.40e6·27-s − 2.38e6·29-s + 6.10e6·31-s + 2.53e6·33-s − 2.04e6·35-s + 3.21e6·37-s − 1.99e6·39-s − 2.58e7·41-s − 1.94e7·43-s + 2.52e6·45-s + 9.09e6·47-s + 1.03e8·49-s − 3.65e7·51-s + 7.10e7·53-s − 6.17e6·55-s + ⋯ |
| L(s) = 1 | + 0.497·3-s − 0.121·5-s + 1.89·7-s − 0.752·9-s + 0.746·11-s − 0.277·13-s − 0.0607·15-s − 1.51·17-s − 1.24·19-s + 0.941·21-s + 0.0198·23-s − 0.985·25-s − 0.872·27-s − 0.625·29-s + 1.18·31-s + 0.371·33-s − 0.230·35-s + 0.281·37-s − 0.138·39-s − 1.42·41-s − 0.868·43-s + 0.0917·45-s + 0.271·47-s + 2.57·49-s − 0.756·51-s + 1.23·53-s − 0.0910·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 3 | \( 1 - 69.8T + 1.96e4T^{2} \) |
| 5 | \( 1 + 170.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.20e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.62e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 5.23e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.06e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.66e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.38e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.10e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.21e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.58e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.94e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 9.09e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.10e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.54e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 8.93e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.09e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.14e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.00e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.19e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.28e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.06e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.90e8T + 7.60e17T^{2} \) |
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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
−10.37336009518271339636695837151, −8.816576035653095629825573360117, −8.499195677179035008486921122305, −7.44422921043513474586672349620, −6.14411779372452055538657269993, −4.82958869235707139845591414762, −4.02726282499298762892637985530, −2.39101970471462830707125356064, −1.62186397027378756611550895400, 0,
1.62186397027378756611550895400, 2.39101970471462830707125356064, 4.02726282499298762892637985530, 4.82958869235707139845591414762, 6.14411779372452055538657269993, 7.44422921043513474586672349620, 8.499195677179035008486921122305, 8.816576035653095629825573360117, 10.37336009518271339636695837151
