| L(s) = 1 | − 5.33·2-s + 8.84·3-s + 20.4·4-s + 3.71·5-s − 47.1·6-s − 66.5·8-s + 51.1·9-s − 19.8·10-s − 5.05·11-s + 180.·12-s + 52.7·13-s + 32.8·15-s + 191.·16-s − 80.7·17-s − 273.·18-s − 59.5·19-s + 75.9·20-s + 26.9·22-s − 83.1·23-s − 588.·24-s − 111.·25-s − 281.·26-s + 213.·27-s + 131.·29-s − 175.·30-s + 146.·31-s − 487.·32-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 1.70·3-s + 2.55·4-s + 0.331·5-s − 3.21·6-s − 2.93·8-s + 1.89·9-s − 0.626·10-s − 0.138·11-s + 4.35·12-s + 1.12·13-s + 0.564·15-s + 2.98·16-s − 1.15·17-s − 3.57·18-s − 0.719·19-s + 0.849·20-s + 0.261·22-s − 0.753·23-s − 5.00·24-s − 0.889·25-s − 2.12·26-s + 1.52·27-s + 0.839·29-s − 1.06·30-s + 0.851·31-s − 2.69·32-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
| \(L(2)\) |
\(\approx\) |
\(1.982772960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.982772960\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5.33T + 8T^{2} \) |
| 3 | \( 1 - 8.84T + 27T^{2} \) |
| 5 | \( 1 - 3.71T + 125T^{2} \) |
| 11 | \( 1 + 5.05T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 80.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 59.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 83.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 131.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 152.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 442.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 488.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 238.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 374.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 903.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 550.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 497.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 465.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.21e3T + 9.12e5T^{2} \) |
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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
−8.698566996168966559028016540512, −8.124271759989126317302389753234, −7.67050807325568928459902617850, −6.61438940986237398940553683869, −6.14334278471463087238667294093, −4.30453175181719421656604115878, −3.29525967116003761208367039915, −2.30559881040134995265431192857, −1.91755165176721645311630810081, −0.76085367606785450918641773753,
0.76085367606785450918641773753, 1.91755165176721645311630810081, 2.30559881040134995265431192857, 3.29525967116003761208367039915, 4.30453175181719421656604115878, 6.14334278471463087238667294093, 6.61438940986237398940553683869, 7.67050807325568928459902617850, 8.124271759989126317302389753234, 8.698566996168966559028016540512
