L(s) = 1 | − i·2-s − 4-s + (−2 + i)5-s − i·7-s + i·8-s + (1 + 2i)10-s + 4i·13-s − 14-s + 16-s + 2i·17-s + 8·19-s + (2 − i)20-s + 8i·23-s + (3 − 4i)25-s + 4·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.894 + 0.447i)5-s − 0.377i·7-s + 0.353i·8-s + (0.316 + 0.632i)10-s + 1.10i·13-s − 0.267·14-s + 0.250·16-s + 0.485i·17-s + 1.83·19-s + (0.447 − 0.223i)20-s + 1.66i·23-s + (0.600 − 0.800i)25-s + 0.784·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00663 + 0.237634i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00663 + 0.237634i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−10.91946341175581805790662520964, −9.750325669363806175511469432261, −9.231004043744810732276820062365, −7.86094990335840108523802619017, −7.40815005074694681976714668290, −6.16337119095468751719743091885, −4.83849137185205450889737786480, −3.85889533379040075872733995812, −3.06440711733884952843546864526, −1.40245221211779527951862810289,
0.63892303427725341813276550352, 2.91353021490250363503403738067, 4.09237732048846523750102007496, 5.16243260990912172191610139147, 5.86477934218199323320095027219, 7.29784820732422506511104865715, 7.70978173658112462198926068231, 8.710471852531880161530895307943, 9.375143355219890940722532391734, 10.49839725824576295199357472012