L(s) = 1 | − 2-s − i·3-s + 4-s + (2 − i)5-s + i·6-s + 3i·7-s − 8-s − 9-s + (−2 + i)10-s − 3·11-s − i·12-s − 4·13-s − 3i·14-s + (−1 − 2i)15-s + 16-s + 7·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.5·4-s + (0.894 − 0.447i)5-s + 0.408i·6-s + 1.13i·7-s − 0.353·8-s − 0.333·9-s + (−0.632 + 0.316i)10-s − 0.904·11-s − 0.288i·12-s − 1.10·13-s − 0.801i·14-s + (−0.258 − 0.516i)15-s + 0.250·16-s + 1.69·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.270573610\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.270573610\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| 37 | \( 1 + (-6 + i)T \) |
good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 + 5iT - 31T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 5iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−9.464422765222231864240239895120, −9.142063290605765694918676228180, −7.944800691313305294392262098999, −7.56616110415167812477128210763, −6.27035422248702251553430672134, −5.62898898145511302419375314104, −4.88171201275687955830839733343, −2.70870410307633660129651567510, −2.36715974832215047004468296495, −0.824332873162489645089156544143,
1.17622194661047059541889486780, 2.67749981244277732549187936917, 3.51741779286829605121937274790, 5.01645746251138040909617075311, 5.61658549998341156981163708296, 6.92438416930772307260789120583, 7.43135631203755985745287464329, 8.338064904277218959335980431427, 9.523444578773864700398708753408, 9.944785342426537184050889714096