| 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 + 2·11-s + i·12-s + i·13-s + 14-s + (−2 − i)15-s + 16-s − 6i·17-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 + 0.603·11-s + 0.288i·12-s + 0.277i·13-s + 0.267·14-s + (−0.516 − 0.258i)15-s + 0.250·16-s − 1.45i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.288406457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288406457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 - iT \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.531142515807889993437623038391, −7.77583520220441225220240888486, −7.02984348567264484331543614934, −6.33964035792722363663016379621, −5.64277173317613014982654692741, −4.69949095770989629480558338137, −4.12620434640993342783958627088, −2.69831473288623403381088195562, −1.50298514028487786538192038318, −0.41565209580724932017156317177,
1.59044428749358217622363911890, 2.49095017041483857660671872142, 3.44637159425141179847749184898, 4.06362679315222992103264443764, 5.10478704832969915447591968417, 6.06893298151005146003160435369, 6.48778858882141430400552780184, 7.75618301813588221475528841216, 8.503877688392302835722420781037, 9.271406667374528854909424332697
