| L(s) = 1 | + 3.46·5-s − 3.46·7-s − 6·17-s − 4·19-s − 6.92·23-s + 6.99·25-s − 3.46·29-s − 3.46·31-s − 11.9·35-s + 6.92·37-s − 6·41-s − 4·43-s + 6.92·47-s + 4.99·49-s − 3.46·53-s − 12·59-s − 6.92·61-s + 4·67-s + 6.92·71-s − 2·73-s + 10.3·79-s − 20.7·85-s + 6·89-s − 13.8·95-s − 2·97-s − 3.46·101-s + 17.3·103-s + ⋯ |
| L(s) = 1 | + 1.54·5-s − 1.30·7-s − 1.45·17-s − 0.917·19-s − 1.44·23-s + 1.39·25-s − 0.643·29-s − 0.622·31-s − 2.02·35-s + 1.13·37-s − 0.937·41-s − 0.609·43-s + 1.01·47-s + 0.714·49-s − 0.475·53-s − 1.56·59-s − 0.887·61-s + 0.488·67-s + 0.822·71-s − 0.234·73-s + 1.16·79-s − 2.25·85-s + 0.635·89-s − 1.42·95-s − 0.203·97-s − 0.344·101-s + 1.70·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.015531523845900068320152927652, −7.86852777929663488987876925219, −6.68571996650659968785279685501, −6.32027506245078211169784598159, −5.73825499223907028216052868382, −4.63929477064860348917448634603, −3.63535021906343364664894789563, −2.49260455118983694588862785334, −1.85410503708777745297468497660, 0,
1.85410503708777745297468497660, 2.49260455118983694588862785334, 3.63535021906343364664894789563, 4.63929477064860348917448634603, 5.73825499223907028216052868382, 6.32027506245078211169784598159, 6.68571996650659968785279685501, 7.86852777929663488987876925219, 9.015531523845900068320152927652
