| L(s) = 1 | + 2·11-s + 13-s + 17-s + 4·19-s − 7·23-s − 29-s + 3·31-s + 6·37-s − 3·41-s − 43-s − 12·47-s + 11·53-s + 3·59-s − 5·61-s − 12·67-s + 4·71-s − 14·73-s + 2·79-s − 3·83-s + 10·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.603·11-s + 0.277·13-s + 0.242·17-s + 0.917·19-s − 1.45·23-s − 0.185·29-s + 0.538·31-s + 0.986·37-s − 0.468·41-s − 0.152·43-s − 1.75·47-s + 1.51·53-s + 0.390·59-s − 0.640·61-s − 1.46·67-s + 0.474·71-s − 1.63·73-s + 0.225·79-s − 0.329·83-s + 1.05·89-s − 1.01·97-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
| \(L(1)\) |
\(\approx\) |
\(2.362168100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.362168100\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.15545663133374, −12.79845500192476, −12.04056094539926, −11.74988544839698, −11.52799157706553, −10.82520930742632, −10.19668070011229, −9.898171172813094, −9.499599511496568, −8.729856123424844, −8.559746607081210, −7.695890646392130, −7.603802427345697, −6.843565994161347, −6.271812394682860, −5.948555413057905, −5.371082501028054, −4.699805784463029, −4.251605999335744, −3.612513080574253, −3.191362294666237, −2.483093748304667, −1.764936495376970, −1.242578102825098, −0.4592380420475426,
0.4592380420475426, 1.242578102825098, 1.764936495376970, 2.483093748304667, 3.191362294666237, 3.612513080574253, 4.251605999335744, 4.699805784463029, 5.371082501028054, 5.948555413057905, 6.271812394682860, 6.843565994161347, 7.603802427345697, 7.695890646392130, 8.559746607081210, 8.729856123424844, 9.499599511496568, 9.898171172813094, 10.19668070011229, 10.82520930742632, 11.52799157706553, 11.74988544839698, 12.04056094539926, 12.79845500192476, 13.15545663133374
