| L(s) = 1 | − 0.364i·2-s + i·3-s + 1.86·4-s + 0.364·6-s + 2.24i·7-s − 1.40i·8-s − 9-s + 4.63·11-s + 1.86i·12-s − 3.75i·13-s + 0.818·14-s + 3.22·16-s − 5.36i·17-s + 0.364i·18-s + 5.66·19-s + ⋯ |
| L(s) = 1 | − 0.257i·2-s + 0.577i·3-s + 0.933·4-s + 0.148·6-s + 0.850i·7-s − 0.497i·8-s − 0.333·9-s + 1.39·11-s + 0.539i·12-s − 1.04i·13-s + 0.218·14-s + 0.805·16-s − 1.30i·17-s + 0.0858i·18-s + 1.29·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.490151648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.490151648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.364iT - 2T^{2} \) |
| 7 | \( 1 - 2.24iT - 7T^{2} \) |
| 11 | \( 1 - 4.63T + 11T^{2} \) |
| 13 | \( 1 + 3.75iT - 13T^{2} \) |
| 17 | \( 1 + 5.36iT - 17T^{2} \) |
| 19 | \( 1 - 5.66T + 19T^{2} \) |
| 23 | \( 1 + 1.32iT - 23T^{2} \) |
| 29 | \( 1 + 8.86T + 29T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 + 2.15iT - 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 + 1.16iT - 43T^{2} \) |
| 47 | \( 1 + 2.36iT - 47T^{2} \) |
| 53 | \( 1 - 14.1iT - 53T^{2} \) |
| 59 | \( 1 + 0.367T + 59T^{2} \) |
| 61 | \( 1 - 5.29T + 61T^{2} \) |
| 67 | \( 1 - 8.06iT - 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 13.7iT - 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 14.4iT - 97T^{2} \) |
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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
−9.448959825814812467739241196942, −8.619613398206562960110742942025, −7.52930624340494866618786339786, −6.94487420005387834012096427469, −5.78123288161131497830656559181, −5.45397909664807751335074839669, −4.09197169462938133812802894420, −3.17944978097171444086361822378, −2.46342957814811443332175040354, −1.08026122698912381706378493240,
1.27371985288315977376527630359, 1.93056586100111985965224411116, 3.41827015891755945928261687521, 4.08211108868191197297051514515, 5.46025989217756168536207487493, 6.33329114121678778515885225116, 6.85192045291598464694118250463, 7.46638609688256412492692365757, 8.214285591949615254719926668595, 9.223317749838877939191188174965
