| L(s) = 1 | + 2-s + 4-s − 5-s + 1.92i·7-s + 8-s − 10-s − 2.15i·11-s + 2.99·13-s + 1.92i·14-s + 16-s − 7.71i·17-s − 4.37i·19-s − 20-s − 2.15i·22-s + 6.55i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.729i·7-s + 0.353·8-s − 0.316·10-s − 0.649i·11-s + 0.830·13-s + 0.515i·14-s + 0.250·16-s − 1.87i·17-s − 1.00i·19-s − 0.223·20-s − 0.459i·22-s + 1.36i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.598923137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.598923137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + (-6.55 + 0.244i)T \) |
| good | 7 | \( 1 - 1.92iT - 7T^{2} \) |
| 11 | \( 1 + 2.15iT - 11T^{2} \) |
| 13 | \( 1 - 2.99T + 13T^{2} \) |
| 17 | \( 1 + 7.71iT - 17T^{2} \) |
| 19 | \( 1 + 4.37iT - 19T^{2} \) |
| 23 | \( 1 - 6.55iT - 23T^{2} \) |
| 29 | \( 1 + 7.88T + 29T^{2} \) |
| 31 | \( 1 - 0.611T + 31T^{2} \) |
| 37 | \( 1 + 6.19iT - 37T^{2} \) |
| 41 | \( 1 + 0.159iT - 41T^{2} \) |
| 47 | \( 1 + 7.35iT - 47T^{2} \) |
| 53 | \( 1 + 3.69iT - 53T^{2} \) |
| 59 | \( 1 - 6.73iT - 59T^{2} \) |
| 61 | \( 1 - 2.00iT - 61T^{2} \) |
| 67 | \( 1 + 3.37T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 1.50iT - 73T^{2} \) |
| 79 | \( 1 - 2.90T + 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 + 3.11T + 89T^{2} \) |
| 97 | \( 1 + 3.54T + 97T^{2} \) |
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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
−8.388268332009071524403685639170, −7.34938664604756258306293403782, −7.06047481404053765025372971427, −5.79409810573027538522899164079, −5.54058468498633236152313783500, −4.60927278514668435813096615185, −3.65667106539362425529870650179, −3.02043568730348878213641339256, −2.08480268015294771185428354038, −0.63085174346524731062025545057,
1.19774267390643267280660756588, 2.15919245859567717313263198908, 3.47421736629459193980868238725, 3.99747293859000309159486678930, 4.55037242069338416701584436916, 5.69581975044640295368001329294, 6.31052434963157469217526106593, 6.99146001413956871179530221483, 7.950174087088016959394049691246, 8.229526610494065110703743270133
