| 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
−10.49839725824576295199357472012, −9.375143355219890940722532391734, −8.710471852531880161530895307943, −7.70978173658112462198926068231, −7.29784820732422506511104865715, −5.86477934218199323320095027219, −5.16243260990912172191610139147, −4.09237732048846523750102007496, −2.91353021490250363503403738067, −0.63892303427725341813276550352,
1.40245221211779527951862810289, 3.06440711733884952843546864526, 3.85889533379040075872733995812, 4.83849137185205450889737786480, 6.16337119095468751719743091885, 7.40815005074694681976714668290, 7.86094990335840108523802619017, 9.231004043744810732276820062365, 9.750325669363806175511469432261, 10.91946341175581805790662520964
