| L(s) = 1 | − 3·3-s + 18.5·5-s − 9.32·7-s + 9·9-s − 39.7·11-s − 32.9·13-s − 55.6·15-s + 90.5·17-s + 72.5·19-s + 27.9·21-s − 45.3·23-s + 218.·25-s − 27·27-s + 143.·29-s + 90.4·31-s + 119.·33-s − 172.·35-s + 1.77·37-s + 98.8·39-s − 195.·41-s + 407.·43-s + 166.·45-s − 278.·47-s − 256.·49-s − 271.·51-s + 241.·53-s − 736.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.65·5-s − 0.503·7-s + 0.333·9-s − 1.08·11-s − 0.703·13-s − 0.957·15-s + 1.29·17-s + 0.876·19-s + 0.290·21-s − 0.411·23-s + 1.75·25-s − 0.192·27-s + 0.918·29-s + 0.524·31-s + 0.628·33-s − 0.835·35-s + 0.00790·37-s + 0.405·39-s − 0.745·41-s + 1.44·43-s + 0.552·45-s − 0.864·47-s − 0.746·49-s − 0.746·51-s + 0.625·53-s − 1.80·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.064508736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064508736\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| good | 5 | \( 1 - 18.5T + 125T^{2} \) |
| 7 | \( 1 + 9.32T + 343T^{2} \) |
| 11 | \( 1 + 39.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 90.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 72.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 143.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 90.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 1.77T + 5.06e4T^{2} \) |
| 41 | \( 1 + 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 407.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 278.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 241.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 149.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 508.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 950.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 803.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 449.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 157.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 175.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 127.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 158.T + 9.12e5T^{2} \) |
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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
−10.00080264256539951008672719313, −9.499302661985459601715556409743, −8.182682420777268319346150049685, −7.17073039326504919991089327435, −6.21968548789269336018119489509, −5.50165826814179012108764238138, −4.91747911024426052054833237430, −3.15805125946845498511731590510, −2.18511239912335008107570510626, −0.842167293582934822875999038399,
0.842167293582934822875999038399, 2.18511239912335008107570510626, 3.15805125946845498511731590510, 4.91747911024426052054833237430, 5.50165826814179012108764238138, 6.21968548789269336018119489509, 7.17073039326504919991089327435, 8.182682420777268319346150049685, 9.499302661985459601715556409743, 10.00080264256539951008672719313
