| L(s) = 1 | − 5.45·2-s + 21.7·4-s + 19.3·5-s − 74.6·8-s − 105.·10-s + 11.2·11-s − 46.3·13-s + 233.·16-s − 97.6·17-s − 98.9·19-s + 419.·20-s − 61.1·22-s + 138.·23-s + 249.·25-s + 252.·26-s + 180.·29-s − 31.9·31-s − 674.·32-s + 532.·34-s − 205.·37-s + 539.·38-s − 1.44e3·40-s + 234.·41-s − 320.·43-s + 243.·44-s − 753.·46-s − 312.·47-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.71·4-s + 1.73·5-s − 3.30·8-s − 3.33·10-s + 0.307·11-s − 0.988·13-s + 3.64·16-s − 1.39·17-s − 1.19·19-s + 4.69·20-s − 0.592·22-s + 1.25·23-s + 1.99·25-s + 1.90·26-s + 1.15·29-s − 0.184·31-s − 3.72·32-s + 2.68·34-s − 0.912·37-s + 2.30·38-s − 5.71·40-s + 0.892·41-s − 1.13·43-s + 0.833·44-s − 2.41·46-s − 0.970·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.45T + 8T^{2} \) |
| 5 | \( 1 - 19.3T + 125T^{2} \) |
| 11 | \( 1 - 11.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 97.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 98.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 180.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 31.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 205.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 320.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 312.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 53.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 400.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 97.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 257.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 253.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 889.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 647.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 673.T + 9.12e5T^{2} \) |
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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
−8.790614805654211236500847970872, −8.597445918451943660605573523551, −7.12201984342116235284839305267, −6.70544992009211824384127699757, −5.98956692708127004478806518884, −4.84147748973002035578153263653, −2.79263034457667444553670450622, −2.16421325523284409120256249426, −1.34102181639747164538667425451, 0,
1.34102181639747164538667425451, 2.16421325523284409120256249426, 2.79263034457667444553670450622, 4.84147748973002035578153263653, 5.98956692708127004478806518884, 6.70544992009211824384127699757, 7.12201984342116235284839305267, 8.597445918451943660605573523551, 8.790614805654211236500847970872
