| L(s) = 1 | − 1.35i·2-s + i·3-s + 0.175·4-s + 1.35·6-s + 1.59i·7-s − 2.93i·8-s − 9-s + 3.33·11-s + 0.175i·12-s − 7.05i·13-s + 2.15·14-s − 3.61·16-s + 4.09i·17-s + 1.35i·18-s − 0.567·19-s + ⋯ |
| L(s) = 1 | − 0.955i·2-s + 0.577i·3-s + 0.0876·4-s + 0.551·6-s + 0.603i·7-s − 1.03i·8-s − 0.333·9-s + 1.00·11-s + 0.0505i·12-s − 1.95i·13-s + 0.576·14-s − 0.904·16-s + 0.993i·17-s + 0.318i·18-s − 0.130·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.979615825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.979615825\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.35iT - 2T^{2} \) |
| 7 | \( 1 - 1.59iT - 7T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 13 | \( 1 + 7.05iT - 13T^{2} \) |
| 17 | \( 1 - 4.09iT - 17T^{2} \) |
| 19 | \( 1 + 0.567T + 19T^{2} \) |
| 23 | \( 1 + 6.30iT - 23T^{2} \) |
| 29 | \( 1 - 2.78T + 29T^{2} \) |
| 31 | \( 1 + 0.995T + 31T^{2} \) |
| 37 | \( 1 + 3.55iT - 37T^{2} \) |
| 41 | \( 1 - 1.16T + 41T^{2} \) |
| 43 | \( 1 + 0.117iT - 43T^{2} \) |
| 47 | \( 1 - 7.64iT - 47T^{2} \) |
| 53 | \( 1 + 0.523iT - 53T^{2} \) |
| 59 | \( 1 + 0.983T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 15.2iT - 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 5.55iT - 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 5.02iT - 83T^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 + 1.70iT - 97T^{2} \) |
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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
−9.165908467421846716421183739581, −8.505282529587938366090700387310, −7.61329833345139645878993070259, −6.39350877290311411923543469006, −5.88825131827140096776033616722, −4.73494438106049336049991126008, −3.76869123397645818015388070172, −3.05710937677810684330817788090, −2.14252392040638561178439772324, −0.77292375439385250853680887899,
1.33119185094530616526675077304, 2.32990165591786468444763733618, 3.73747981969190567247440688064, 4.66681288047890567726900911108, 5.66509269576773327568590942079, 6.60631641531887825451821308406, 6.97449561012641665622103032021, 7.49047789774529967142255188398, 8.563912775627038110916302015580, 9.138121659690971372981937794936
