L(s) = 1 | + (−0.965 − 0.258i)2-s + (−0.5 − 0.866i)3-s + (0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (0.258 + 0.965i)6-s + (−0.366 − 1.36i)7-s + (−0.707 − 0.707i)8-s + (−0.499 + 0.866i)9-s + (−0.866 + 0.500i)10-s + (−0.965 − 0.258i)11-s − i·12-s − i·13-s + 1.41i·14-s + (−0.965 − 0.258i)15-s + (0.500 + 0.866i)16-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (−0.5 − 0.866i)3-s + (0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (0.258 + 0.965i)6-s + (−0.366 − 1.36i)7-s + (−0.707 − 0.707i)8-s + (−0.499 + 0.866i)9-s + (−0.866 + 0.500i)10-s + (−0.965 − 0.258i)11-s − i·12-s − i·13-s + 1.41i·14-s + (−0.965 − 0.258i)15-s + (0.500 + 0.866i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4224253616\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4224253616\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + iT - T^{2} \) |
| 37 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
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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
−8.174332278476691813758049921923, −7.82363405040287164039747627991, −7.12119996957822516223429903735, −6.23426466023452328970251866172, −5.67704992427796142777817454456, −4.63479030314768775125890668045, −3.33349697394925499315382527545, −2.34953382139590935956635215608, −1.29752099452932883122082804271, −0.37470022237946577085710324937,
2.02094518189893321835550899041, 2.59385368366083507030341953558, 3.70134487541304804005621733216, 5.23300403087651287128768967900, 5.54181517284575613570493564243, 6.38020679974929825263718973600, 6.87744296137170326584886229911, 8.000986704678052552512506677815, 8.868820827975447006958200812021, 9.344241160518247309204511584872