| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s − 4·7-s + 8-s + 9-s − 3·10-s + 3·11-s + 12-s + 2·13-s − 4·14-s − 3·15-s + 16-s + 18-s + 8·19-s − 3·20-s − 4·21-s + 3·22-s + 6·23-s + 24-s + 4·25-s + 2·26-s + 27-s − 4·28-s + 3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.904·11-s + 0.288·12-s + 0.554·13-s − 1.06·14-s − 0.774·15-s + 1/4·16-s + 0.235·18-s + 1.83·19-s − 0.670·20-s − 0.872·21-s + 0.639·22-s + 1.25·23-s + 0.204·24-s + 4/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.469111763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.469111763\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.359378456870067145421721931455, −8.497024114195934312365354568876, −7.51544900682975926506183651656, −6.98640743251474618531181839040, −6.22795400541615145935265131281, −5.09608703225063482365487698332, −3.92195504246663866758478973809, −3.52194656344149573570504842676, −2.84456835613246057654603603677, −0.981399216701231737472054832458,
0.981399216701231737472054832458, 2.84456835613246057654603603677, 3.52194656344149573570504842676, 3.92195504246663866758478973809, 5.09608703225063482365487698332, 6.22795400541615145935265131281, 6.98640743251474618531181839040, 7.51544900682975926506183651656, 8.497024114195934312365354568876, 9.359378456870067145421721931455
