L(s) = 1 | + 3-s − 2·5-s + 9-s − 11-s + 6·13-s − 2·15-s − 2·17-s − 4·19-s − 25-s + 27-s + 2·29-s + 8·31-s − 33-s − 6·37-s + 6·39-s − 10·41-s − 4·43-s − 2·45-s − 8·47-s − 2·51-s − 6·53-s + 2·55-s − 4·57-s − 4·59-s − 10·61-s − 12·65-s − 12·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.174·33-s − 0.986·37-s + 0.960·39-s − 1.56·41-s − 0.609·43-s − 0.298·45-s − 1.16·47-s − 0.280·51-s − 0.824·53-s + 0.269·55-s − 0.529·57-s − 0.520·59-s − 1.28·61-s − 1.48·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.244985955\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.244985955\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.69091381502370, −13.37159837966752, −12.78871783573565, −12.27055554345833, −11.73490295919921, −11.27112494438054, −10.85933179262444, −10.21047126081086, −9.942458744968332, −9.029496540137397, −8.593043034638283, −8.321321937366810, −7.949308285660674, −7.205909849042764, −6.677148188398076, −6.240064939915363, −5.653232505049799, −4.681471861709781, −4.456542779335204, −3.814263967702939, −3.178690617075192, −2.911672390630634, −1.760737574148101, −1.483347830642525, −0.3280024187788447,
0.3280024187788447, 1.483347830642525, 1.760737574148101, 2.911672390630634, 3.178690617075192, 3.814263967702939, 4.456542779335204, 4.681471861709781, 5.653232505049799, 6.240064939915363, 6.677148188398076, 7.205909849042764, 7.949308285660674, 8.321321937366810, 8.593043034638283, 9.029496540137397, 9.942458744968332, 10.21047126081086, 10.85933179262444, 11.27112494438054, 11.73490295919921, 12.27055554345833, 12.78871783573565, 13.37159837966752, 13.69091381502370