| L(s) = 1 | − 0.732i·2-s + 1.46·4-s + 5-s − 4.08·7-s − 2.53i·8-s − 0.732i·10-s + 2.45i·11-s − 0.393·13-s + 2.99i·14-s + 1.07·16-s − 2.58i·17-s − 7.38i·19-s + 1.46·20-s + 1.79·22-s + 8.05·23-s + ⋯ |
| L(s) = 1 | − 0.517i·2-s + 0.731·4-s + 0.447·5-s − 1.54·7-s − 0.896i·8-s − 0.231i·10-s + 0.740i·11-s − 0.109·13-s + 0.799i·14-s + 0.267·16-s − 0.628i·17-s − 1.69i·19-s + 0.327·20-s + 0.383·22-s + 1.67·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.719985435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719985435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + (-0.825 + 5.32i)T \) |
| good | 2 | \( 1 + 0.732iT - 2T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 11 | \( 1 - 2.45iT - 11T^{2} \) |
| 13 | \( 1 + 0.393T + 13T^{2} \) |
| 17 | \( 1 + 2.58iT - 17T^{2} \) |
| 19 | \( 1 + 7.38iT - 19T^{2} \) |
| 23 | \( 1 - 8.05T + 23T^{2} \) |
| 31 | \( 1 + 6.86iT - 31T^{2} \) |
| 37 | \( 1 + 8.71iT - 37T^{2} \) |
| 41 | \( 1 - 11.3iT - 41T^{2} \) |
| 43 | \( 1 + 7.75iT - 43T^{2} \) |
| 47 | \( 1 + 2.06iT - 47T^{2} \) |
| 53 | \( 1 + 3.02T + 53T^{2} \) |
| 59 | \( 1 - 8.03T + 59T^{2} \) |
| 61 | \( 1 - 11.5iT - 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 + 3.80iT - 73T^{2} \) |
| 79 | \( 1 - 7.38iT - 79T^{2} \) |
| 83 | \( 1 + 8.43T + 83T^{2} \) |
| 89 | \( 1 - 6.04iT - 89T^{2} \) |
| 97 | \( 1 - 10.4iT - 97T^{2} \) |
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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
−9.518649817223412378513928579767, −9.082264060389849909019464430446, −7.47901574242176415544377797289, −6.89014073842796200127741007420, −6.33443463475071133525070083813, −5.24434405197204878582601416943, −4.03575300019802805463445373810, −2.85036291793179715963463685917, −2.42706393255930914624360446320, −0.70768930642772335663164903114,
1.47709398145327119030735771265, 2.95675921724181644586067679350, 3.46179980494234207438018244444, 5.15112651467284266367453777806, 5.96459481464839014727902435072, 6.52861698888283191122571772798, 7.15570090125386358277303086338, 8.264126946983208700689253540416, 8.948407311346068778315179193818, 9.957401537626835680406315964724
