| L(s) = 1 | + 30.3·3-s + 25·5-s + 49·7-s + 677.·9-s + 604.·11-s + 383.·13-s + 758.·15-s − 791.·17-s − 2.07e3·19-s + 1.48e3·21-s − 2.52e3·23-s + 625·25-s + 1.31e4·27-s + 372.·29-s − 2.48e3·31-s + 1.83e4·33-s + 1.22e3·35-s − 6.55e3·37-s + 1.16e4·39-s − 1.61e4·41-s + 1.54e4·43-s + 1.69e4·45-s + 2.22e4·47-s + 2.40e3·49-s − 2.40e4·51-s − 2.62e4·53-s + 1.51e4·55-s + ⋯ |
| L(s) = 1 | + 1.94·3-s + 0.447·5-s + 0.377·7-s + 2.78·9-s + 1.50·11-s + 0.630·13-s + 0.870·15-s − 0.664·17-s − 1.32·19-s + 0.735·21-s − 0.994·23-s + 0.200·25-s + 3.47·27-s + 0.0821·29-s − 0.464·31-s + 2.93·33-s + 0.169·35-s − 0.786·37-s + 1.22·39-s − 1.50·41-s + 1.27·43-s + 1.24·45-s + 1.47·47-s + 0.142·49-s − 1.29·51-s − 1.28·53-s + 0.673·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.322876041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.322876041\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 30.3T + 243T^{2} \) |
| 11 | \( 1 - 604.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 383.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 791.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.07e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.52e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 372.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.48e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.61e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.54e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.22e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.62e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.89e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.79e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.98e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.25e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.20e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.49e4T + 8.58e9T^{2} \) |
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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
−10.76258148432783031401690249236, −9.774863763247863647977465474549, −8.811141341444328614954385961373, −8.543856633598979092115045294914, −7.25277371116993730539449923949, −6.31271445615688204044001848628, −4.35418940734519868825688185379, −3.66708966895661366146336163250, −2.24429078518721282654680591133, −1.47507092495482398870471603134,
1.47507092495482398870471603134, 2.24429078518721282654680591133, 3.66708966895661366146336163250, 4.35418940734519868825688185379, 6.31271445615688204044001848628, 7.25277371116993730539449923949, 8.543856633598979092115045294914, 8.811141341444328614954385961373, 9.774863763247863647977465474549, 10.76258148432783031401690249236
