| L(s) = 1 | + 5·7-s − 11-s + 13-s + 3·17-s − 6·19-s − 3·23-s + 4·29-s − 5·37-s − 11·41-s − 6·43-s + 18·49-s + 9·53-s − 12·59-s − 5·61-s − 8·67-s + 13·71-s + 10·73-s − 5·77-s + 17·79-s − 4·83-s + 11·89-s + 5·91-s + 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.88·7-s − 0.301·11-s + 0.277·13-s + 0.727·17-s − 1.37·19-s − 0.625·23-s + 0.742·29-s − 0.821·37-s − 1.71·41-s − 0.914·43-s + 18/7·49-s + 1.23·53-s − 1.56·59-s − 0.640·61-s − 0.977·67-s + 1.54·71-s + 1.17·73-s − 0.569·77-s + 1.91·79-s − 0.439·83-s + 1.16·89-s + 0.524·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.054187838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.054187838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.31798206779211, −12.45009394593850, −12.03177284267928, −11.91889392773554, −11.14756933731778, −10.77287940639612, −10.41330722039614, −10.01391684221416, −9.202646019350542, −8.655417271135273, −8.350297150466480, −7.894522672406011, −7.600761502606775, −6.744422950100312, −6.448034236904194, −5.663482686774201, −5.223985894190465, −4.774117705123680, −4.345410707940901, −3.677961721988995, −3.134073866938987, −2.152956351394744, −1.939564468470661, −1.296625247590221, −0.4994665200011066,
0.4994665200011066, 1.296625247590221, 1.939564468470661, 2.152956351394744, 3.134073866938987, 3.677961721988995, 4.345410707940901, 4.774117705123680, 5.223985894190465, 5.663482686774201, 6.448034236904194, 6.744422950100312, 7.600761502606775, 7.894522672406011, 8.350297150466480, 8.655417271135273, 9.202646019350542, 10.01391684221416, 10.41330722039614, 10.77287940639612, 11.14756933731778, 11.91889392773554, 12.03177284267928, 12.45009394593850, 13.31798206779211
