| L(s) = 1 | + (1.61 + 1.17i)3-s + (−2.14 + 6.61i)5-s + (1.55 + 2.14i)7-s + (−1.54 − 4.75i)9-s + (5.86 + 9.30i)11-s + (−5.91 + 1.92i)13-s + (−11.2 + 8.17i)15-s + (−0.865 − 0.281i)17-s + (−11.3 + 15.5i)19-s + 5.29i·21-s − 38.1·23-s + (−18.8 − 13.7i)25-s + (8.65 − 26.6i)27-s + (13.1 + 18.0i)29-s + (12.8 + 39.4i)31-s + ⋯ |
| L(s) = 1 | + (0.539 + 0.391i)3-s + (−0.429 + 1.32i)5-s + (0.222 + 0.305i)7-s + (−0.171 − 0.528i)9-s + (0.533 + 0.845i)11-s + (−0.454 + 0.147i)13-s + (−0.749 + 0.544i)15-s + (−0.0509 − 0.0165i)17-s + (−0.595 + 0.819i)19-s + 0.251i·21-s − 1.66·23-s + (−0.755 − 0.548i)25-s + (0.320 − 0.986i)27-s + (0.451 + 0.621i)29-s + (0.413 + 1.27i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.833i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.551 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.714810 + 1.33022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.714810 + 1.33022i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.55 - 2.14i)T \) |
| 11 | \( 1 + (-5.86 - 9.30i)T \) |
| good | 3 | \( 1 + (-1.61 - 1.17i)T + (2.78 + 8.55i)T^{2} \) |
| 5 | \( 1 + (2.14 - 6.61i)T + (-20.2 - 14.6i)T^{2} \) |
| 13 | \( 1 + (5.91 - 1.92i)T + (136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (0.865 + 0.281i)T + (233. + 169. i)T^{2} \) |
| 19 | \( 1 + (11.3 - 15.5i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + 38.1T + 529T^{2} \) |
| 29 | \( 1 + (-13.1 - 18.0i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-12.8 - 39.4i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-32.3 + 23.5i)T + (423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (28.6 - 39.4i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + 81.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-59.8 - 43.4i)T + (682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-26.4 - 81.3i)T + (-2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-62.2 + 45.2i)T + (1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (70.6 + 22.9i)T + (3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + 12.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-13.6 + 41.8i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-33.9 - 46.7i)T + (-1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-57.8 + 18.7i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-74.2 - 24.1i)T + (5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 - 42.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-0.260 - 0.801i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.97102703668444708970229094166, −10.68245460309398131528833924501, −10.01621563808279128742876617523, −9.030982111095893600583687390319, −7.974909912500900218939600912776, −6.97042374537557936756367922722, −6.09289052641876349671584433824, −4.36885964804768738506295910643, −3.43991611233588835003971709500, −2.23477917709520432983636066081,
0.67700434642782588845012501123, 2.27701521671627017303441768157, 3.98359843062012709191255773288, 4.92691477647652673073734667402, 6.20234819428203825954663319270, 7.67495286366398058944908079116, 8.276692622698564223976307648584, 8.965959120301938738234212519994, 10.13696160417895933798236880787, 11.40587743230814290140314351771
