L(s) = 1 | − i·2-s + i·3-s + 4-s + (−1 − 2i)5-s + 6-s − 3i·8-s − 9-s + (−2 + i)10-s − 6·11-s + i·12-s − 2i·13-s + (2 − i)15-s − 16-s − 4i·17-s + i·18-s − 6·19-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 − 1.06i·8-s − 0.333·9-s + (−0.632 + 0.316i)10-s − 1.80·11-s + 0.288i·12-s − 0.554i·13-s + (0.516 − 0.258i)15-s − 0.250·16-s − 0.970i·17-s + 0.235i·18-s − 1.37·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.231278 - 0.979709i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.231278 - 0.979709i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 16iT - 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−10.24954674872127833581824633607, −9.390474647614821434448921376422, −8.262007300807147454324838074952, −7.71382622968980540644197573214, −6.39823248084765288028849059082, −5.20980453827603346495185123115, −4.49572321492708365576392047484, −3.23140721448490455171248779770, −2.32101591337753869525714945934, −0.46777483556197831922160617880,
2.15426545704181043667133261772, 2.92714915453075174170310452548, 4.50967170295167665734040054949, 5.79385320903557292475169219793, 6.50068627462598728544158889301, 7.18107025616131508097129846225, 8.118093345649354850343782386613, 8.390750672063355757797823404983, 10.26264449642343933639969614141, 10.65432352920307798340698525594