| L(s) = 1 | − 2.96i·5-s + 0.934·7-s + 5.98i·11-s + 10.1·13-s + 7.86i·17-s + 4.58·19-s + 35.6i·23-s + 16.1·25-s − 9.37i·29-s − 54.2·31-s − 2.77i·35-s + 18.4·37-s − 10.8i·41-s − 55.8·43-s + 72.9i·47-s + ⋯ |
| L(s) = 1 | − 0.593i·5-s + 0.133·7-s + 0.544i·11-s + 0.780·13-s + 0.462i·17-s + 0.241·19-s + 1.55i·23-s + 0.647·25-s − 0.323i·29-s − 1.75·31-s − 0.0792i·35-s + 0.498·37-s − 0.264i·41-s − 1.29·43-s + 1.55i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.472563159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472563159\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.96iT - 25T^{2} \) |
| 7 | \( 1 - 0.934T + 49T^{2} \) |
| 11 | \( 1 - 5.98iT - 121T^{2} \) |
| 13 | \( 1 - 10.1T + 169T^{2} \) |
| 17 | \( 1 - 7.86iT - 289T^{2} \) |
| 19 | \( 1 - 4.58T + 361T^{2} \) |
| 23 | \( 1 - 35.6iT - 529T^{2} \) |
| 29 | \( 1 + 9.37iT - 841T^{2} \) |
| 31 | \( 1 + 54.2T + 961T^{2} \) |
| 37 | \( 1 - 18.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 10.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 72.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 44.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 82.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 62.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 34.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 70.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 136.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 44.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 91.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 99.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 27.7T + 9.40e3T^{2} \) |
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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
−8.750393025463228139898870994393, −8.072140960386394112542934052853, −7.33913917927422077715571611835, −6.50391095701953469633843143265, −5.58481800151668505852524546958, −4.99720372181950437684643376933, −4.00963245660393926866958533154, −3.29222697129176802964669164373, −1.92313641966403096647042920446, −1.15759886503789552021192629263,
0.34852653834339562505079269574, 1.61160693135225333335893590886, 2.79263812620270730108903281498, 3.46310241201994926217256542469, 4.45694486454065696592247901033, 5.37276716175013537594379217116, 6.18188832361418468842651676357, 6.87888776107102823730744676042, 7.55286648342239133740141533392, 8.624466342392265896338772944148
