| L(s) = 1 | + 1.08i·2-s + i·3-s + 0.812·4-s − 1.08·6-s − 3.08i·7-s + 3.06i·8-s − 9-s − 1.14·11-s + 0.812i·12-s − 4.07i·13-s + 3.36·14-s − 1.71·16-s + 4.62i·17-s − 1.08i·18-s + 5.96·19-s + ⋯ |
| L(s) = 1 | + 0.770i·2-s + 0.577i·3-s + 0.406·4-s − 0.444·6-s − 1.16i·7-s + 1.08i·8-s − 0.333·9-s − 0.346·11-s + 0.234i·12-s − 1.13i·13-s + 0.899·14-s − 0.428·16-s + 1.12i·17-s − 0.256i·18-s + 1.36·19-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
| \(L(1)\) |
\(\approx\) |
\(2.079231984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.079231984\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.08iT - 2T^{2} \) |
| 7 | \( 1 + 3.08iT - 7T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 + 4.07iT - 13T^{2} \) |
| 17 | \( 1 - 4.62iT - 17T^{2} \) |
| 19 | \( 1 - 5.96T + 19T^{2} \) |
| 23 | \( 1 - 2.32iT - 23T^{2} \) |
| 29 | \( 1 - 5.28T + 29T^{2} \) |
| 31 | \( 1 + 0.589T + 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 9.49T + 41T^{2} \) |
| 43 | \( 1 + 2.42iT - 43T^{2} \) |
| 47 | \( 1 - 6.04iT - 47T^{2} \) |
| 53 | \( 1 - 3.24iT - 53T^{2} \) |
| 59 | \( 1 + 3.18T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 3.15iT - 67T^{2} \) |
| 71 | \( 1 - 6.46T + 71T^{2} \) |
| 73 | \( 1 + 7.20iT - 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 12.3iT - 83T^{2} \) |
| 89 | \( 1 + 1.08T + 89T^{2} \) |
| 97 | \( 1 - 4.52iT - 97T^{2} \) |
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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
−9.479783472845939049972380165151, −8.256245785667211985664147157131, −7.87287655044973381045725517943, −7.13875898688238483223228074922, −6.22956969688816897687620883755, −5.49082285168640753012797350460, −4.70254263051549981374915195560, −3.60060493074471594357234691280, −2.77100074698927945804718789784, −1.13701142648610746279607878939,
0.898853485848274788196836218819, 2.25663635822703932274388750450, 2.59873796253237360022666581554, 3.75452196553835758988249499693, 5.03445238284068845225097475144, 5.84425014216410106943073952419, 6.77621039651392336564536942761, 7.32104106759194340597507142694, 8.305690584476190053192993015167, 9.328213823631307846910782888044
