L(s) = 1 | + (0.946 − 1.05i)2-s + 3-s + (−0.208 − 1.98i)4-s − 4.42i·5-s + (0.946 − 1.05i)6-s + 2.81·7-s + (−2.28 − 1.66i)8-s + 9-s + (−4.65 − 4.19i)10-s + 4.49·11-s + (−0.208 − 1.98i)12-s + 6.44i·13-s + (2.66 − 2.96i)14-s − 4.42i·15-s + (−3.91 + 0.831i)16-s + 1.33·17-s + ⋯ |
L(s) = 1 | + (0.669 − 0.743i)2-s + 0.577·3-s + (−0.104 − 0.994i)4-s − 1.98i·5-s + (0.386 − 0.429i)6-s + 1.06·7-s + (−0.808 − 0.587i)8-s + 0.333·9-s + (−1.47 − 1.32i)10-s + 1.35·11-s + (−0.0603 − 0.574i)12-s + 1.78i·13-s + (0.712 − 0.791i)14-s − 1.14i·15-s + (−0.978 + 0.207i)16-s + 0.323·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45500 - 2.60437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45500 - 2.60437i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.946 + 1.05i)T \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + (6.48 + 4.99i)T \) |
good | 5 | \( 1 + 4.42iT - 5T^{2} \) |
| 7 | \( 1 - 2.81T + 7T^{2} \) |
| 11 | \( 1 - 4.49T + 11T^{2} \) |
| 13 | \( 1 - 6.44iT - 13T^{2} \) |
| 17 | \( 1 - 1.33T + 17T^{2} \) |
| 19 | \( 1 - 1.70iT - 19T^{2} \) |
| 23 | \( 1 - 3.41iT - 23T^{2} \) |
| 29 | \( 1 + 3.13T + 29T^{2} \) |
| 31 | \( 1 + 8.45T + 31T^{2} \) |
| 37 | \( 1 - 5.75T + 37T^{2} \) |
| 41 | \( 1 - 5.15iT - 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 - 4.52iT - 47T^{2} \) |
| 53 | \( 1 - 4.63iT - 53T^{2} \) |
| 59 | \( 1 + 10.2iT - 59T^{2} \) |
| 61 | \( 1 - 9.66iT - 61T^{2} \) |
| 71 | \( 1 + 0.479iT - 71T^{2} \) |
| 73 | \( 1 - 3.68T + 73T^{2} \) |
| 79 | \( 1 - 9.91T + 79T^{2} \) |
| 83 | \( 1 + 8.53iT - 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 13.2iT - 97T^{2} \) |
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
−9.547579020418817213975991735242, −9.309783335858710618732331760369, −8.629817373884982389913018301986, −7.56413391461962434274013967540, −6.16135023402138816310315312519, −5.12668193266789644319023167626, −4.33833296110290089539309500263, −3.86822167803576465186280195162, −1.70138153561820949620897959108, −1.47307042151190013346026417793,
2.28008976829764549839016074722, 3.29245408312173660757933625689, 3.95985539681378689489408392010, 5.37814419726990218860082220050, 6.28529438596998242806140119160, 7.17754597117143209437493947425, 7.68778022062201706783587906632, 8.516340372453144480775662496835, 9.663910477523645655820984286649, 10.75583204730280342740873743470