| L(s) = 1 | + 3.41i·3-s − 1.54·5-s + 2.64i·7-s − 2.65·9-s + 4.48i·11-s − 1.54·13-s − 5.29i·15-s + 23.6·17-s − 24.8i·19-s − 9.03·21-s + 35.2i·23-s − 22.5·25-s + 21.6i·27-s − 22.4·29-s + 46.7i·31-s + ⋯ |
| L(s) = 1 | + 1.13i·3-s − 0.309·5-s + 0.377i·7-s − 0.295·9-s + 0.407i·11-s − 0.119·13-s − 0.352i·15-s + 1.39·17-s − 1.30i·19-s − 0.430·21-s + 1.53i·23-s − 0.903·25-s + 0.802i·27-s − 0.774·29-s + 1.50i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.294517222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294517222\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 - 3.41iT - 9T^{2} \) |
| 5 | \( 1 + 1.54T + 25T^{2} \) |
| 11 | \( 1 - 4.48iT - 121T^{2} \) |
| 13 | \( 1 + 1.54T + 169T^{2} \) |
| 17 | \( 1 - 23.6T + 289T^{2} \) |
| 19 | \( 1 + 24.8iT - 361T^{2} \) |
| 23 | \( 1 - 35.2iT - 529T^{2} \) |
| 29 | \( 1 + 22.4T + 841T^{2} \) |
| 31 | \( 1 - 46.7iT - 961T^{2} \) |
| 37 | \( 1 - 58.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 26.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 17.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 36.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 97.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 61.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 37.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 33.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 69.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 38.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 3.61iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 44.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 96.1T + 9.40e3T^{2} \) |
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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
−9.589403034756321926266575583317, −8.998802022025881438710498294164, −7.82699305531792088577762140813, −7.34430881053551367976248827932, −6.10511015559386625036657523731, −5.23572535604071187297890468582, −4.60817321739594473483389903838, −3.65710603866805242958524059280, −2.91969252473253601653920045442, −1.42500387626336305613909958200,
0.35731015220892620143535767457, 1.35712191247542349415635776299, 2.42600716547161018002246118532, 3.62020619878021606850755790448, 4.43921756210953651746337085780, 5.87652842502466333367306569151, 6.17226714387798161766273817492, 7.41410310262031145519843996190, 7.74101722123409329713688806181, 8.340906738056490804926941687042
