L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.707 − 1.22i)3-s + (−0.499 + 0.866i)4-s + (2.12 + 3.67i)5-s − 1.41·6-s + 0.999·8-s + (0.500 + 0.866i)9-s + (2.12 − 3.67i)10-s + (0.5 − 0.866i)11-s + (0.707 + 1.22i)12-s + 6·15-s + (−0.5 − 0.866i)16-s + (−2.82 + 4.89i)17-s + (0.499 − 0.866i)18-s − 4.24·20-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.408 − 0.707i)3-s + (−0.249 + 0.433i)4-s + (0.948 + 1.64i)5-s − 0.577·6-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.670 − 1.16i)10-s + (0.150 − 0.261i)11-s + (0.204 + 0.353i)12-s + 1.54·15-s + (−0.125 − 0.216i)16-s + (−0.685 + 1.18i)17-s + (0.117 − 0.204i)18-s − 0.948·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.733347359\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.733347359\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.707 + 1.22i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.12 - 3.67i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (2.82 - 4.89i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (0.707 - 1.22i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-2.12 - 3.67i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4 - 6.92i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.41 + 2.44i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (-4.24 + 7.34i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−10.18627553900618061274281436577, −9.199738636852081754999587505386, −8.230225051663784141190443047227, −7.51291375672964869510772347803, −6.51913572396882816143215733480, −6.13149707476088979412324054824, −4.47590798445976548610636401238, −3.17280420684963810221051575129, −2.41866996381734089683057062447, −1.63731177589728058753183449524,
0.863390962916596040672524380146, 2.20744775122042189013747546161, 3.96766298188808790951595826413, 4.74174954866515649927987989756, 5.46319089850240842925834156012, 6.35847628994209351256829494358, 7.48684758473498329486200643415, 8.475520609925464626455094685063, 9.189411241816531751841551166480, 9.514289404603703577775870179117