L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.5 + 0.866i)3-s + 1.00i·4-s + (−0.965 − 0.258i)5-s + (0.965 − 0.258i)6-s + (−1.22 + 1.22i)7-s + (0.707 − 0.707i)8-s + (−0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (−0.866 − 0.500i)12-s + (−0.707 − 0.707i)13-s + 1.73·14-s + (0.707 − 0.707i)15-s − 1.00·16-s + (−0.366 − 0.366i)17-s + (−0.258 + 0.965i)18-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.5 + 0.866i)3-s + 1.00i·4-s + (−0.965 − 0.258i)5-s + (0.965 − 0.258i)6-s + (−1.22 + 1.22i)7-s + (0.707 − 0.707i)8-s + (−0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (−0.866 − 0.500i)12-s + (−0.707 − 0.707i)13-s + 1.73·14-s + (0.707 − 0.707i)15-s − 1.00·16-s + (−0.366 − 0.366i)17-s + (−0.258 + 0.965i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2508892361\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2508892361\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.965 + 0.258i)T \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 47 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 1.93iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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
−9.492939663466584911317802781285, −8.924381518418183991234583780646, −8.211978454130485024358988711243, −7.16620615272404515806460097240, −6.22176428299551150821333765458, −5.16386838129367475050927464789, −4.25034984568747819746263077057, −3.25239770704115588825129588254, −2.65007174436710436220478710794, −0.33771820937268692910193092819,
0.991281592294065864685394546355, 2.64647866243250958844298617214, 4.05934879015453388052614133463, 4.93472065335988754865192816582, 6.38343384193938593128807318054, 6.62232055153585748126703186344, 7.33610321080108248008391726431, 7.924810111034153616537701601952, 8.784627539648574389151069453813, 9.945188298261589673775880616515