L(s) = 1 | − i·2-s + (1.22 + 1.22i)3-s − 4-s + 2.44·5-s + (1.22 − 1.22i)6-s + (−1 + 2.44i)7-s + i·8-s + 2.99i·9-s − 2.44i·10-s + i·11-s + (−1.22 − 1.22i)12-s + 2.44i·13-s + (2.44 + i)14-s + (2.99 + 2.99i)15-s + 16-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.707 + 0.707i)3-s − 0.5·4-s + 1.09·5-s + (0.499 − 0.499i)6-s + (−0.377 + 0.925i)7-s + 0.353i·8-s + 0.999i·9-s − 0.774i·10-s + 0.301i·11-s + (−0.353 − 0.353i)12-s + 0.679i·13-s + (0.654 + 0.267i)14-s + (0.774 + 0.774i)15-s + 0.250·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.84209 + 0.371303i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.84209 + 0.371303i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.22 - 1.22i)T \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
| 11 | \( 1 - iT \) |
good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.34iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 + 7.34iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−11.09039944615098990883027434050, −9.859905633298188573875088637343, −9.448540034235144596051268695239, −8.975069161766040035312444208117, −7.69737614284267516009526011982, −6.20024566235226822015062428004, −5.22544427313452284854025498179, −4.16924452692247176807868247550, −2.74965798719301971390388905474, −2.08737189456365738072980099548,
1.22025378874221273541587975198, 2.86870169389061181974713472745, 4.11248175776761430513953717730, 5.72753102526927301458850406712, 6.35601509657208608386346382020, 7.33362723222914321537196924773, 8.135424366356313869913742821043, 9.057983506326068758389594068103, 9.976858969392988484306277389827, 10.61559546111206046610654996878