| L(s) = 1 | + 5-s − 6·13-s − 2·17-s + 6·19-s + 23-s + 25-s − 6·29-s − 4·31-s + 4·37-s − 6·41-s + 4·43-s + 4·47-s − 7·49-s − 6·53-s − 14·59-s + 6·61-s − 6·65-s − 4·67-s + 2·71-s + 10·73-s − 14·79-s − 4·83-s − 2·85-s − 18·89-s + 6·95-s + 12·97-s − 18·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.66·13-s − 0.485·17-s + 1.37·19-s + 0.208·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.657·37-s − 0.937·41-s + 0.609·43-s + 0.583·47-s − 49-s − 0.824·53-s − 1.82·59-s + 0.768·61-s − 0.744·65-s − 0.488·67-s + 0.237·71-s + 1.17·73-s − 1.57·79-s − 0.439·83-s − 0.216·85-s − 1.90·89-s + 0.615·95-s + 1.21·97-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.86925416011163500099272666862, −7.36373532159906450745663114896, −6.69471041338947683851255760769, −5.68904757800175628811765528476, −5.14177464600567861541603093738, −4.37308418080693672900604941930, −3.25893603408588238372613411933, −2.47425138183039818206519953316, −1.50641296926337566041149424095, 0,
1.50641296926337566041149424095, 2.47425138183039818206519953316, 3.25893603408588238372613411933, 4.37308418080693672900604941930, 5.14177464600567861541603093738, 5.68904757800175628811765528476, 6.69471041338947683851255760769, 7.36373532159906450745663114896, 7.86925416011163500099272666862
