L(s) = 1 | + (−1.22 + 0.707i)2-s + (2.54 − 1.58i)3-s + (0.999 − 1.73i)4-s + (−5.68 + 3.28i)5-s + (−1.99 + 3.74i)6-s + 2.82i·8-s + (3.98 − 8.07i)9-s + (4.64 − 8.04i)10-s + (0.357 + 0.206i)11-s + (−0.196 − 5.99i)12-s + 20.5·13-s + (−9.29 + 17.3i)15-s + (−2.00 − 3.46i)16-s + (13.7 + 7.93i)17-s + (0.832 + 12.7i)18-s + (8 + 13.8i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.849 − 0.528i)3-s + (0.249 − 0.433i)4-s + (−1.13 + 0.657i)5-s + (−0.333 + 0.623i)6-s + 0.353i·8-s + (0.442 − 0.896i)9-s + (0.464 − 0.804i)10-s + (0.0324 + 0.0187i)11-s + (−0.0163 − 0.499i)12-s + 1.58·13-s + (−0.619 + 1.15i)15-s + (−0.125 − 0.216i)16-s + (0.808 + 0.467i)17-s + (0.0462 + 0.705i)18-s + (0.421 + 0.729i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.45083 + 0.115925i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45083 + 0.115925i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 + (-2.54 + 1.58i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (5.68 - 3.28i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-0.357 - 0.206i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 20.5T + 169T^{2} \) |
| 17 | \( 1 + (-13.7 - 7.93i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-8 - 13.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-31.1 + 18.0i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 20.8iT - 841T^{2} \) |
| 31 | \( 1 + (-2.77 + 4.79i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (10 + 17.3i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 76.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 51.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-7.34 + 4.24i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (44.0 + 25.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (1.42 + 0.824i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-33.4 - 57.9i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (24.7 - 42.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 87.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-6.16 + 10.6i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-42.4 - 73.5i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 4.12iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (27.0 - 15.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 68.8T + 9.40e3T^{2} \) |
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
−11.44289827943726863623081372007, −10.65184753187617256026705246970, −9.470474585267255961254640143510, −8.383708692858983340989952137169, −7.904702025624492626974736864664, −6.97530235793747774548712379573, −6.03426748886586270348351718670, −3.99091994226387915465234989758, −3.02349357292368682893727408174, −1.14885814929620886645444178517,
1.12165303810224317291297010059, 3.12352061763795136720779507763, 3.92181221532003982712100595502, 5.18755010336256427673913783083, 7.14088967575965145511043355351, 7.973770176920776602777964210624, 8.814547347525931158950878343251, 9.319619255301785830873277306945, 10.67655879127716309526401526890, 11.29699108913895061396126731242