| L(s) = 1 | − 142.·5-s − 217. i·7-s + 25.7i·11-s − 1.39e3·13-s − 4.00e3·17-s + 129. i·19-s − 3.69e3i·23-s + 4.75e3·25-s − 4.08e4·29-s − 4.61e4i·31-s + 3.10e4i·35-s + 6.59e4·37-s + 4.18e4·41-s + 3.00e4i·43-s + 7.97e4i·47-s + ⋯ |
| L(s) = 1 | − 1.14·5-s − 0.634i·7-s + 0.0193i·11-s − 0.632·13-s − 0.814·17-s + 0.0188i·19-s − 0.303i·23-s + 0.304·25-s − 1.67·29-s − 1.54i·31-s + 0.724i·35-s + 1.30·37-s + 0.606·41-s + 0.378i·43-s + 0.768i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8607098240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8607098240\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 142.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 217. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 25.7iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.39e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 4.00e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 129. iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 3.69e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 4.08e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 4.61e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 6.59e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 4.18e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 3.00e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 7.97e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.37e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 3.00e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.43e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 4.06e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.49e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.66e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.52e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 5.12e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 7.97e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 8.78e3T + 8.32e11T^{2} \) |
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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
−11.12739253012957309849006669723, −9.973747748913728931383737838967, −8.964288697473439134667833228344, −7.73958147961816564542375687013, −7.32129068608946931980459059845, −5.98103078507131104421576885076, −4.49520933077232282211169073082, −3.86588018919974343743212448040, −2.40526708828014231062312675497, −0.68523633823761009337230993037,
0.32637002900939270770102468578, 2.07466718804535530587511994578, 3.38538128252098027896490840935, 4.46438785781174030124054574420, 5.57244233282508707134345949584, 6.90233331077121564981519544228, 7.75662924272451518515950803554, 8.701982017339126636351518065860, 9.588374411493037647362427272448, 10.88533461631097109146467636885
