| L(s) = 1 | + i·3-s − 5i·7-s − 9-s + 6·11-s − 3i·13-s − 2i·17-s + 19-s + 5·21-s + 2i·23-s − i·27-s + 6·29-s + 3·31-s + 6i·33-s + 6i·37-s + 3·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.88i·7-s − 0.333·9-s + 1.80·11-s − 0.832i·13-s − 0.485i·17-s + 0.229·19-s + 1.09·21-s + 0.417i·23-s − 0.192i·27-s + 1.11·29-s + 0.538·31-s + 1.04i·33-s + 0.986i·37-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.131440817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131440817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 3iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 + 8iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−8.070386165258264988331779367625, −7.45444409110125013809191357255, −6.65214468058777236027927195718, −6.18874262511404438246195101406, −4.89018878615758594764240389838, −4.43643856529949201010740593328, −3.62655754323216977496943798067, −3.10268610706440176264105835414, −1.42464396943373685245798694693, −0.66526899198926756244062975514,
1.21221986866717236464162381646, 2.07188570304636866346138555998, 2.81703981853751883824249755605, 3.90278830921214390507713179810, 4.73140944595475011944813448837, 5.76164771756570215256683818429, 6.26422861346898765050741812747, 6.73642010792931587839690470639, 7.73568343271214195814678494758, 8.584753092487217925910789315590
