| L(s) = 1 | − 1.41·5-s + 4·11-s + 4.24·13-s + 7.07·17-s + 5.65·19-s + 8·23-s − 2.99·25-s − 2·29-s + 4·37-s − 9.89·41-s + 4·43-s + 5.65·47-s − 4·53-s − 5.65·55-s − 11.3·59-s + 1.41·61-s − 6·65-s + 12·67-s − 15.5·73-s + 16·79-s − 5.65·83-s − 10.0·85-s − 7.07·89-s − 8.00·95-s + 7.07·97-s − 12.7·101-s − 5.65·103-s + ⋯ |
| L(s) = 1 | − 0.632·5-s + 1.20·11-s + 1.17·13-s + 1.71·17-s + 1.29·19-s + 1.66·23-s − 0.599·25-s − 0.371·29-s + 0.657·37-s − 1.54·41-s + 0.609·43-s + 0.825·47-s − 0.549·53-s − 0.762·55-s − 1.47·59-s + 0.181·61-s − 0.744·65-s + 1.46·67-s − 1.82·73-s + 1.80·79-s − 0.620·83-s − 1.08·85-s − 0.749·89-s − 0.820·95-s + 0.717·97-s − 1.26·101-s − 0.557·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.476166248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.476166248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−7.86513316277952428779557402296, −7.32808517220458697643926683149, −6.59971232858100171533763595007, −5.80088943061804128712024570899, −5.19940259835401539143184768722, −4.18736436155663171537463701263, −3.50448925254493422864703024220, −3.07155568713705922640635244408, −1.47524322373324930347325096035, −0.922253072603553264704378484969,
0.922253072603553264704378484969, 1.47524322373324930347325096035, 3.07155568713705922640635244408, 3.50448925254493422864703024220, 4.18736436155663171537463701263, 5.19940259835401539143184768722, 5.80088943061804128712024570899, 6.59971232858100171533763595007, 7.32808517220458697643926683149, 7.86513316277952428779557402296
