| L(s) = 1 | + 2·7-s − 2·11-s − 13-s + 2·17-s − 4·19-s + 4·29-s − 8·31-s + 6·37-s + 6·41-s − 4·43-s + 8·47-s − 3·49-s − 2·53-s − 10·59-s + 14·61-s + 16·67-s − 4·71-s − 8·73-s − 4·77-s + 8·79-s + 12·83-s − 6·89-s − 2·91-s − 12·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.603·11-s − 0.277·13-s + 0.485·17-s − 0.917·19-s + 0.742·29-s − 1.43·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.274·53-s − 1.30·59-s + 1.79·61-s + 1.95·67-s − 0.474·71-s − 0.936·73-s − 0.455·77-s + 0.900·79-s + 1.31·83-s − 0.635·89-s − 0.209·91-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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
| \(L(1)\) |
\(\approx\) |
\(2.356224175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.356224175\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.98984598122707, −12.66424145475778, −12.31903082198050, −11.60461360728041, −11.24501352944614, −10.78128850683479, −10.39925137306480, −9.822948699157504, −9.347568564106178, −8.815220380173942, −8.231599686335149, −7.932122814025349, −7.441451767534238, −6.895624421387608, −6.330058617791344, −5.697808128778615, −5.341193205085550, −4.699241898765632, −4.330036806723996, −3.662761199184319, −3.051060523641975, −2.313475001461312, −1.999870216765581, −1.145107646657868, −0.4626002771603013,
0.4626002771603013, 1.145107646657868, 1.999870216765581, 2.313475001461312, 3.051060523641975, 3.662761199184319, 4.330036806723996, 4.699241898765632, 5.341193205085550, 5.697808128778615, 6.330058617791344, 6.895624421387608, 7.441451767534238, 7.932122814025349, 8.231599686335149, 8.815220380173942, 9.347568564106178, 9.822948699157504, 10.39925137306480, 10.78128850683479, 11.24501352944614, 11.60461360728041, 12.31903082198050, 12.66424145475778, 12.98984598122707
