| L(s) = 1 | + 1.75·2-s − 3.28·3-s + 1.07·4-s − 1.87·5-s − 5.76·6-s − 1.62·8-s + 7.81·9-s − 3.29·10-s − 2.70·11-s − 3.53·12-s − 3.71·13-s + 6.17·15-s − 4.99·16-s + 2.05·17-s + 13.6·18-s − 7.60·19-s − 2.01·20-s − 4.75·22-s + 4.81·23-s + 5.33·24-s − 1.47·25-s − 6.51·26-s − 15.8·27-s − 8.01·29-s + 10.8·30-s + 0.938·31-s − 5.50·32-s + ⋯ |
| L(s) = 1 | + 1.23·2-s − 1.89·3-s + 0.536·4-s − 0.839·5-s − 2.35·6-s − 0.574·8-s + 2.60·9-s − 1.04·10-s − 0.817·11-s − 1.01·12-s − 1.03·13-s + 1.59·15-s − 1.24·16-s + 0.497·17-s + 3.22·18-s − 1.74·19-s − 0.450·20-s − 1.01·22-s + 1.00·23-s + 1.08·24-s − 0.294·25-s − 1.27·26-s − 3.04·27-s − 1.48·29-s + 1.97·30-s + 0.168·31-s − 0.973·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1746826962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1746826962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 1.75T + 2T^{2} \) |
| 3 | \( 1 + 3.28T + 3T^{2} \) |
| 5 | \( 1 + 1.87T + 5T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 13 | \( 1 + 3.71T + 13T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 + 7.60T + 19T^{2} \) |
| 23 | \( 1 - 4.81T + 23T^{2} \) |
| 29 | \( 1 + 8.01T + 29T^{2} \) |
| 31 | \( 1 - 0.938T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 + 7.94T + 41T^{2} \) |
| 43 | \( 1 + 5.32T + 43T^{2} \) |
| 47 | \( 1 + 7.32T + 47T^{2} \) |
| 53 | \( 1 + 8.15T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 3.70T + 71T^{2} \) |
| 73 | \( 1 + 2.98T + 73T^{2} \) |
| 79 | \( 1 + 0.365T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 5.66T + 89T^{2} \) |
| 97 | \( 1 - 2.08T + 97T^{2} \) |
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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
−7.63749833380897445832246006650, −7.07612546598672491997632736429, −6.36771314079293290377891076726, −5.73268599594836384435286025052, −5.07664366166557046482345114386, −4.64248506039168072570987779643, −4.04707295008649650249278287177, −3.11067590715979983510134378861, −1.86095991363397838986371308341, −0.18944376381956469792569650514,
0.18944376381956469792569650514, 1.86095991363397838986371308341, 3.11067590715979983510134378861, 4.04707295008649650249278287177, 4.64248506039168072570987779643, 5.07664366166557046482345114386, 5.73268599594836384435286025052, 6.36771314079293290377891076726, 7.07612546598672491997632736429, 7.63749833380897445832246006650
