| L(s) = 1 | + 2-s + 3-s + 4-s − 2.80·5-s + 6-s − 4.03·7-s + 8-s + 9-s − 2.80·10-s − 0.386·11-s + 12-s − 0.0552·13-s − 4.03·14-s − 2.80·15-s + 16-s − 1.34·17-s + 18-s − 19-s − 2.80·20-s − 4.03·21-s − 0.386·22-s − 2.17·23-s + 24-s + 2.85·25-s − 0.0552·26-s + 27-s − 4.03·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.25·5-s + 0.408·6-s − 1.52·7-s + 0.353·8-s + 0.333·9-s − 0.886·10-s − 0.116·11-s + 0.288·12-s − 0.0153·13-s − 1.07·14-s − 0.723·15-s + 0.250·16-s − 0.326·17-s + 0.235·18-s − 0.229·19-s − 0.626·20-s − 0.880·21-s − 0.0824·22-s − 0.453·23-s + 0.204·24-s + 0.570·25-s − 0.0108·26-s + 0.192·27-s − 0.762·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.010410554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.010410554\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 2.80T + 5T^{2} \) |
| 7 | \( 1 + 4.03T + 7T^{2} \) |
| 11 | \( 1 + 0.386T + 11T^{2} \) |
| 13 | \( 1 + 0.0552T + 13T^{2} \) |
| 17 | \( 1 + 1.34T + 17T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 29 | \( 1 + 3.62T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + 0.635T + 41T^{2} \) |
| 43 | \( 1 + 0.308T + 43T^{2} \) |
| 47 | \( 1 + 0.542T + 47T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 3.60T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 4.73T + 73T^{2} \) |
| 79 | \( 1 - 6.82T + 79T^{2} \) |
| 83 | \( 1 - 3.48T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
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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
−7.948093119123166776411585959761, −7.37003246098599163007886427348, −6.58642005956396437145863734156, −6.13855494033189697162529624412, −5.03130919021517862616427782722, −4.11511020050065659199128950537, −3.73514352290254875058773859859, −2.99955591168534674836546034778, −2.26808368006076189776690349321, −0.61819255987411233985208614741,
0.61819255987411233985208614741, 2.26808368006076189776690349321, 2.99955591168534674836546034778, 3.73514352290254875058773859859, 4.11511020050065659199128950537, 5.03130919021517862616427782722, 6.13855494033189697162529624412, 6.58642005956396437145863734156, 7.37003246098599163007886427348, 7.948093119123166776411585959761
