L(s) = 1 | + 2·13-s + 6·17-s − 4·19-s + 6·29-s − 4·31-s − 2·37-s + 6·41-s + 8·43-s − 12·47-s + 6·53-s + 12·59-s − 2·61-s + 8·67-s + 14·73-s + 16·79-s + 12·83-s + 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 0.824·53-s + 1.56·59-s − 0.256·61-s + 0.977·67-s + 1.63·73-s + 1.80·79-s + 1.31·83-s + 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.422630537\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.422630537\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.04650523793161, −12.68223974161961, −12.36678256386288, −11.73337585803760, −11.34192117467100, −10.71921527549252, −10.43645320728373, −9.863144958068950, −9.423539838474794, −8.830525050169621, −8.402644855485250, −7.871615258004236, −7.522634809403516, −6.799729495598001, −6.321078953820848, −5.956162002950151, −5.187702186839038, −4.957899751289151, −4.096950467472133, −3.658124828265495, −3.215411802430124, −2.373431367680353, −1.961542205898623, −1.011027920890447, −0.6351287585130339,
0.6351287585130339, 1.011027920890447, 1.961542205898623, 2.373431367680353, 3.215411802430124, 3.658124828265495, 4.096950467472133, 4.957899751289151, 5.187702186839038, 5.956162002950151, 6.321078953820848, 6.799729495598001, 7.522634809403516, 7.871615258004236, 8.402644855485250, 8.830525050169621, 9.423539838474794, 9.863144958068950, 10.43645320728373, 10.71921527549252, 11.34192117467100, 11.73337585803760, 12.36678256386288, 12.68223974161961, 13.04650523793161