| L(s) = 1 | − 5.03·3-s + 54.3·5-s − 110.·7-s − 217.·9-s + 385.·11-s + 286.·13-s − 273.·15-s + 472.·17-s − 2.66e3·19-s + 557.·21-s + 1.80e3·23-s − 176.·25-s + 2.31e3·27-s + 2.93e3·29-s − 595.·31-s − 1.94e3·33-s − 6.01e3·35-s + 9.51e3·37-s − 1.44e3·39-s − 1.15e4·41-s + 6.18e3·43-s − 1.18e4·45-s + 1.77e4·47-s − 4.55e3·49-s − 2.37e3·51-s − 8.69e3·53-s + 2.09e4·55-s + ⋯ |
| L(s) = 1 | − 0.323·3-s + 0.971·5-s − 0.853·7-s − 0.895·9-s + 0.961·11-s + 0.469·13-s − 0.313·15-s + 0.396·17-s − 1.69·19-s + 0.275·21-s + 0.712·23-s − 0.0563·25-s + 0.612·27-s + 0.647·29-s − 0.111·31-s − 0.310·33-s − 0.829·35-s + 1.14·37-s − 0.151·39-s − 1.07·41-s + 0.510·43-s − 0.870·45-s + 1.17·47-s − 0.270·49-s − 0.128·51-s − 0.425·53-s + 0.934·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 5.03T + 243T^{2} \) |
| 5 | \( 1 - 54.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 110.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 385.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 286.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 472.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.66e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 595.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.15e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.18e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.77e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.69e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.64e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.75e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.30e4T + 8.58e9T^{2} \) |
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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
−9.005786701848196464229199895568, −8.115493223425465756291592562346, −6.67445136237315096810236161971, −6.27998816763890258433199273276, −5.62361847636925119465040005153, −4.42868867339728115350730736127, −3.32432736212616561947657672820, −2.36170830511598305489153354171, −1.19218309129788927872451086377, 0,
1.19218309129788927872451086377, 2.36170830511598305489153354171, 3.32432736212616561947657672820, 4.42868867339728115350730736127, 5.62361847636925119465040005153, 6.27998816763890258433199273276, 6.67445136237315096810236161971, 8.115493223425465756291592562346, 9.005786701848196464229199895568
