| L(s) = 1 | + (1.73 − i)2-s + (1.99 − 3.46i)4-s − 6.92·7-s − 7.99i·8-s − 6.92i·11-s + 2i·13-s + (−11.9 + 6.92i)14-s + (−8 − 13.8i)16-s + 10i·17-s − 20.7i·19-s + (−6.92 − 11.9i)22-s − 27.7·23-s + (2 + 3.46i)26-s + (−13.8 + 23.9i)28-s − 26·29-s + ⋯ |
| L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s − 0.989·7-s − 0.999i·8-s − 0.629i·11-s + 0.153i·13-s + (−0.857 + 0.494i)14-s + (−0.5 − 0.866i)16-s + 0.588i·17-s − 1.09i·19-s + (−0.314 − 0.545i)22-s − 1.20·23-s + (0.0769 + 0.133i)26-s + (−0.494 + 0.857i)28-s − 0.896·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0599i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.382573276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382573276\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.73 + i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 6.92T + 49T^{2} \) |
| 11 | \( 1 + 6.92iT - 121T^{2} \) |
| 13 | \( 1 - 2iT - 169T^{2} \) |
| 17 | \( 1 - 10iT - 289T^{2} \) |
| 19 | \( 1 + 20.7iT - 361T^{2} \) |
| 23 | \( 1 + 27.7T + 529T^{2} \) |
| 29 | \( 1 + 26T + 841T^{2} \) |
| 31 | \( 1 - 6.92iT - 961T^{2} \) |
| 37 | \( 1 + 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 58T + 1.68e3T^{2} \) |
| 43 | \( 1 - 48.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 69.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 74iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 90.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26T + 3.72e3T^{2} \) |
| 67 | \( 1 - 6.92T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 82T + 7.92e3T^{2} \) |
| 97 | \( 1 + 2iT - 9.40e3T^{2} \) |
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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
−9.651388874594476198520487953952, −8.898493321068531807978420474311, −7.61769363588504025026169046108, −6.53799204133396654300458880770, −6.02998921227114954906906025667, −4.98004958583509001448077390301, −3.85637789603428651709051236909, −3.13459602729980929185372989468, −1.93808391154810705231376904164, −0.30232173369696205475541553934,
2.03326989321967257497185454359, 3.25583913829660784592173995770, 4.03639584380994739206567476718, 5.15498754846501660428307704866, 6.03708457365699409013626165805, 6.77057692349209971531797298310, 7.62055089871512385832723311028, 8.429006397809141385896667747655, 9.624587433620819467830966536777, 10.20239056604255985688924708157
