| L(s) = 1 | + 2-s − 0.551·3-s + 4-s − 5-s − 0.551·6-s + 3.69·7-s + 8-s − 2.69·9-s − 10-s − 1.44·11-s − 0.551·12-s − 5.69·13-s + 3.69·14-s + 0.551·15-s + 16-s − 4.55·17-s − 2.69·18-s + 6.79·19-s − 20-s − 2.03·21-s − 1.44·22-s − 0.551·24-s + 25-s − 5.69·26-s + 3.14·27-s + 3.69·28-s + 10.4·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.318·3-s + 0.5·4-s − 0.447·5-s − 0.225·6-s + 1.39·7-s + 0.353·8-s − 0.898·9-s − 0.316·10-s − 0.436·11-s − 0.159·12-s − 1.57·13-s + 0.987·14-s + 0.142·15-s + 0.250·16-s − 1.10·17-s − 0.635·18-s + 1.55·19-s − 0.223·20-s − 0.445·21-s − 0.308·22-s − 0.112·24-s + 0.200·25-s − 1.11·26-s + 0.604·27-s + 0.698·28-s + 1.94·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.551T + 3T^{2} \) |
| 7 | \( 1 - 3.69T + 7T^{2} \) |
| 11 | \( 1 + 1.44T + 11T^{2} \) |
| 13 | \( 1 + 5.69T + 13T^{2} \) |
| 17 | \( 1 + 4.55T + 17T^{2} \) |
| 19 | \( 1 - 6.79T + 19T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 2.59T + 31T^{2} \) |
| 37 | \( 1 + 0.207T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 4.49T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 3.10T + 67T^{2} \) |
| 71 | \( 1 + 4.55T + 71T^{2} \) |
| 73 | \( 1 - 2.49T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 7.59T + 83T^{2} \) |
| 89 | \( 1 - 7.39T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.895337797287901256158958447114, −7.07813616535049958922285038028, −6.37808478109772162192608334779, −5.25787114561030642735006405671, −4.96789176554309532420247198927, −4.49000056674169312895275447553, −3.16795434838478300119454357167, −2.58464313245200089056499169978, −1.49995593021040348811320882232, 0,
1.49995593021040348811320882232, 2.58464313245200089056499169978, 3.16795434838478300119454357167, 4.49000056674169312895275447553, 4.96789176554309532420247198927, 5.25787114561030642735006405671, 6.37808478109772162192608334779, 7.07813616535049958922285038028, 7.895337797287901256158958447114
