| L(s) = 1 | − 3.81·2-s − 8.81·3-s + 6.59·4-s + 14.5·5-s + 33.6·6-s + 5.38·8-s + 50.6·9-s − 55.6·10-s + 26.7·11-s − 58.0·12-s + 15.0·13-s − 128.·15-s − 73.2·16-s + 5.84·17-s − 193.·18-s + 35.8·19-s + 95.9·20-s − 102.·22-s + 32.7·23-s − 47.4·24-s + 87.0·25-s − 57.5·26-s − 208.·27-s − 160.·29-s + 490.·30-s − 77.8·31-s + 236.·32-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 1.69·3-s + 0.823·4-s + 1.30·5-s + 2.29·6-s + 0.237·8-s + 1.87·9-s − 1.75·10-s + 0.731·11-s − 1.39·12-s + 0.321·13-s − 2.20·15-s − 1.14·16-s + 0.0833·17-s − 2.53·18-s + 0.432·19-s + 1.07·20-s − 0.988·22-s + 0.297·23-s − 0.403·24-s + 0.696·25-s − 0.434·26-s − 1.48·27-s − 1.02·29-s + 2.98·30-s − 0.451·31-s + 1.30·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8251124353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8251124353\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 + 3.81T + 8T^{2} \) |
| 3 | \( 1 + 8.81T + 27T^{2} \) |
| 5 | \( 1 - 14.5T + 125T^{2} \) |
| 11 | \( 1 - 26.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.84T + 4.91e3T^{2} \) |
| 19 | \( 1 - 35.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 32.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 77.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 423.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 6.86T + 6.89e4T^{2} \) |
| 43 | \( 1 + 371.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 150.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 661.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 554.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 326.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 351.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 223.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 354.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 637.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 431.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 783.T + 9.12e5T^{2} \) |
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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
−8.945803730235181800423942293887, −7.891122294165126076458909400748, −6.92777117744282661840146055158, −6.45762925804900302644933973629, −5.65826850546780008112300348068, −5.05675684191279362162413903244, −3.94740850991897403152933279283, −2.16343094444216545895203524897, −1.30222222586862586823488369920, −0.63463435253952221238747931549,
0.63463435253952221238747931549, 1.30222222586862586823488369920, 2.16343094444216545895203524897, 3.94740850991897403152933279283, 5.05675684191279362162413903244, 5.65826850546780008112300348068, 6.45762925804900302644933973629, 6.92777117744282661840146055158, 7.891122294165126076458909400748, 8.945803730235181800423942293887
