| L(s) = 1 | + 19.6·5-s + 19.6·11-s + 54·13-s + 39.2·17-s − 2·19-s + 196.·23-s + 259.·25-s + 98.1·29-s − 201·31-s − 202·37-s − 470.·41-s − 244·43-s + 235.·47-s + 647.·53-s + 384.·55-s + 608.·59-s + 588·61-s + 1.05e3·65-s − 302·67-s − 156.·71-s + 446·73-s − 267·79-s − 725.·83-s + 769.·85-s + 117.·89-s − 39.2·95-s − 595·97-s + ⋯ |
| L(s) = 1 | + 1.75·5-s + 0.537·11-s + 1.15·13-s + 0.559·17-s − 0.0241·19-s + 1.77·23-s + 2.07·25-s + 0.628·29-s − 1.16·31-s − 0.897·37-s − 1.79·41-s − 0.865·43-s + 0.730·47-s + 1.67·53-s + 0.943·55-s + 1.34·59-s + 1.23·61-s + 2.02·65-s − 0.550·67-s − 0.262·71-s + 0.715·73-s − 0.380·79-s − 0.960·83-s + 0.982·85-s + 0.140·89-s − 0.0423·95-s − 0.622·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.964215480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.964215480\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 19.6T + 125T^{2} \) |
| 11 | \( 1 - 19.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 196.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 98.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 201T + 2.97e4T^{2} \) |
| 37 | \( 1 + 202T + 5.06e4T^{2} \) |
| 41 | \( 1 + 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 244T + 7.95e4T^{2} \) |
| 47 | \( 1 - 235.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 647.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 608.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 588T + 2.26e5T^{2} \) |
| 67 | \( 1 + 302T + 3.00e5T^{2} \) |
| 71 | \( 1 + 156.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 446T + 3.89e5T^{2} \) |
| 79 | \( 1 + 267T + 4.93e5T^{2} \) |
| 83 | \( 1 + 725.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 117.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 595T + 9.12e5T^{2} \) |
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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
−8.870149107048123352840541687699, −8.545219305561585976393953223822, −6.97871178973125800396754433976, −6.63199857316967915236907191484, −5.54342882285696911851858016211, −5.24317458482210161900916835175, −3.81626891204726869422986486217, −2.86604894553626255750391243953, −1.73344126364117142192173170123, −1.04520667818275745770923256936,
1.04520667818275745770923256936, 1.73344126364117142192173170123, 2.86604894553626255750391243953, 3.81626891204726869422986486217, 5.24317458482210161900916835175, 5.54342882285696911851858016211, 6.63199857316967915236907191484, 6.97871178973125800396754433976, 8.545219305561585976393953223822, 8.870149107048123352840541687699
