L(s) = 1 | − i·2-s + i·3-s − 4-s + (−1 + 2i)5-s + 6-s − i·7-s + i·8-s − 9-s + (2 + i)10-s − 5·11-s − i·12-s − 4i·13-s − 14-s + (−2 − i)15-s + 16-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.447 + 0.894i)5-s + 0.408·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s + (0.632 + 0.316i)10-s − 1.50·11-s − 0.288i·12-s − 1.10i·13-s − 0.267·14-s + (−0.516 − 0.258i)15-s + 0.250·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.388722 - 0.628966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.388722 - 0.628966i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 13iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−10.18874160642155211997753089142, −9.192122096228278683362482097397, −7.953132079078275551480790536103, −7.64378437401921216055901242366, −6.22374225769867074162870602737, −5.19178200871947861699615301855, −4.29577267265008237659604437346, −3.12450224960058915117254787891, −2.63243592550045901050677890798, −0.36339773528994289038468446489,
1.40172750430398820767725177859, 2.98857294562736104201925937535, 4.36839168767893088398497180253, 5.30221655571935217398086244202, 5.87805932560752718461395752671, 7.29121855227043982553249199146, 7.61616000176016779642244096715, 8.561884869855235754382523425160, 9.211098191991994807640338601733, 10.10600486064744863422335040363