| L(s) = 1 | − 8.00e3·3-s + 3.90e5·5-s − 2.54e7·7-s − 6.50e7·9-s + 2.81e8·11-s − 1.52e9·13-s − 3.12e9·15-s + 5.46e10·17-s − 6.88e8·19-s + 2.03e11·21-s − 3.91e11·23-s + 1.52e11·25-s + 1.55e12·27-s + 5.12e12·29-s + 7.31e10·31-s − 2.24e12·33-s − 9.94e12·35-s − 6.81e12·37-s + 1.22e13·39-s − 5.76e13·41-s − 7.57e13·43-s − 2.54e13·45-s + 4.60e13·47-s + 4.16e14·49-s − 4.37e14·51-s + 6.58e14·53-s + 1.09e14·55-s + ⋯ |
| L(s) = 1 | − 0.704·3-s + 0.447·5-s − 1.66·7-s − 0.503·9-s + 0.395·11-s − 0.518·13-s − 0.314·15-s + 1.90·17-s − 0.00930·19-s + 1.17·21-s − 1.04·23-s + 0.200·25-s + 1.05·27-s + 1.90·29-s + 0.0153·31-s − 0.278·33-s − 0.746·35-s − 0.319·37-s + 0.365·39-s − 1.12·41-s − 0.987·43-s − 0.225·45-s + 0.282·47-s + 1.78·49-s − 1.33·51-s + 1.45·53-s + 0.176·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 3.90e5T \) |
| good | 3 | \( 1 + 8.00e3T + 1.29e8T^{2} \) |
| 7 | \( 1 + 2.54e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 2.81e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.52e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 5.46e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 6.88e8T + 5.48e21T^{2} \) |
| 23 | \( 1 + 3.91e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 5.12e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 7.31e10T + 2.25e25T^{2} \) |
| 37 | \( 1 + 6.81e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 5.76e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 7.57e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 4.60e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 6.58e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 2.98e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 8.50e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 6.12e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 5.41e14T + 2.96e31T^{2} \) |
| 73 | \( 1 + 7.16e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 5.45e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 3.64e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 6.81e14T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.20e17T + 5.95e33T^{2} \) |
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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
−10.20159809148455284314896040870, −9.869304902286570624100124741669, −8.466791654189739562683208560282, −6.89169930888481419973888658788, −6.10168002837486496971293166847, −5.25198775759465458230814010785, −3.59196442053614487099116156424, −2.65898612860599523421170307885, −0.990827976280117132610330401199, 0,
0.990827976280117132610330401199, 2.65898612860599523421170307885, 3.59196442053614487099116156424, 5.25198775759465458230814010785, 6.10168002837486496971293166847, 6.89169930888481419973888658788, 8.466791654189739562683208560282, 9.869304902286570624100124741669, 10.20159809148455284314896040870
