| L(s) = 1 | − 24.9·3-s + 64·4-s + 622.·7-s − 106.·9-s − 2.54e3·11-s − 1.59e3·12-s + 4.09e3·16-s + 9.79e3·17-s − 1.55e4·21-s + 8.06e3·23-s + 1.56e4·25-s + 2.08e4·27-s + 3.98e4·28-s − 3.55e4·29-s + 1.70e4·31-s + 6.34e4·33-s − 6.81e3·36-s + 8.17e4·37-s + 5.04e3·41-s − 1.62e5·44-s − 1.02e5·48-s + 2.69e5·49-s − 2.44e5·51-s + 4.10e5·59-s − 3.29e5·61-s − 6.62e4·63-s + 2.62e5·64-s + ⋯ |
| L(s) = 1 | − 0.924·3-s + 4-s + 1.81·7-s − 0.146·9-s − 1.90·11-s − 0.924·12-s + 16-s + 1.99·17-s − 1.67·21-s + 0.662·23-s + 25-s + 1.05·27-s + 1.81·28-s − 1.45·29-s + 0.572·31-s + 1.76·33-s − 0.146·36-s + 1.61·37-s + 0.0731·41-s − 1.90·44-s − 0.924·48-s + 2.29·49-s − 1.84·51-s + 1.99·59-s − 1.45·61-s − 0.265·63-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.985158584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.985158584\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 83 | \( 1 + 5.71e5T \) |
| good | 2 | \( 1 - 64T^{2} \) |
| 3 | \( 1 + 24.9T + 729T^{2} \) |
| 5 | \( 1 - 1.56e4T^{2} \) |
| 7 | \( 1 - 622.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 2.54e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 4.82e6T^{2} \) |
| 17 | \( 1 - 9.79e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 4.70e7T^{2} \) |
| 23 | \( 1 - 8.06e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 3.55e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 1.70e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 8.17e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 5.04e3T + 4.75e9T^{2} \) |
| 43 | \( 1 - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.21e10T^{2} \) |
| 59 | \( 1 - 4.10e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 3.29e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.43e11T^{2} \) |
| 89 | \( 1 - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.32e11T^{2} \) |
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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
−12.78657430564750147975665733133, −11.71427153756419330143876730757, −11.03910578201710557491080057597, −10.34691407711791203056918432090, −8.121910208453279042605445104595, −7.47694947242684595425563892875, −5.67054099502640540395492377036, −5.10193024848142208756142631992, −2.68681851751014706715160076360, −1.08991546679160834032766900168,
1.08991546679160834032766900168, 2.68681851751014706715160076360, 5.10193024848142208756142631992, 5.67054099502640540395492377036, 7.47694947242684595425563892875, 8.121910208453279042605445104595, 10.34691407711791203056918432090, 11.03910578201710557491080057597, 11.71427153756419330143876730757, 12.78657430564750147975665733133
