| L(s) = 1 | − 3-s + 9-s − 2·11-s − 13-s − 2·17-s − 5·19-s − 6·23-s − 27-s + 10·29-s − 3·31-s + 2·33-s + 2·37-s + 39-s + 8·41-s − 43-s + 2·47-s + 2·51-s − 4·53-s + 5·57-s − 10·59-s − 7·61-s + 3·67-s + 6·69-s + 8·71-s + 14·73-s + 81-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.485·17-s − 1.14·19-s − 1.25·23-s − 0.192·27-s + 1.85·29-s − 0.538·31-s + 0.348·33-s + 0.328·37-s + 0.160·39-s + 1.24·41-s − 0.152·43-s + 0.291·47-s + 0.280·51-s − 0.549·53-s + 0.662·57-s − 1.30·59-s − 0.896·61-s + 0.366·67-s + 0.722·69-s + 0.949·71-s + 1.63·73-s + 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7951654040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7951654040\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−14.24633273176779, −13.84210576581781, −13.31420803943913, −12.65640800256996, −12.28783570445660, −12.00586529123309, −11.02297986228085, −10.87658124872038, −10.41201658674525, −9.675858975616674, −9.377383067021891, −8.514365315279121, −8.074409783297952, −7.656085737045349, −6.783358932957887, −6.471834070289756, −5.890400484911014, −5.313855797683433, −4.546696209958043, −4.331877887861100, −3.501044208860813, −2.593220998616282, −2.194490057761582, −1.278506681802521, −0.3241867691525944,
0.3241867691525944, 1.278506681802521, 2.194490057761582, 2.593220998616282, 3.501044208860813, 4.331877887861100, 4.546696209958043, 5.313855797683433, 5.890400484911014, 6.471834070289756, 6.783358932957887, 7.656085737045349, 8.074409783297952, 8.514365315279121, 9.377383067021891, 9.675858975616674, 10.41201658674525, 10.87658124872038, 11.02297986228085, 12.00586529123309, 12.28783570445660, 12.65640800256996, 13.31420803943913, 13.84210576581781, 14.24633273176779
