L(s) = 1 | + 3-s + 4·7-s + 9-s − 11-s + 6·13-s − 2·17-s + 4·21-s + 23-s − 5·25-s + 27-s + 4·31-s − 33-s + 2·37-s + 6·39-s + 4·41-s − 4·43-s + 9·49-s − 2·51-s + 8·53-s − 4·59-s + 2·61-s + 4·63-s − 10·67-s + 69-s + 16·71-s − 10·73-s − 5·75-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.485·17-s + 0.872·21-s + 0.208·23-s − 25-s + 0.192·27-s + 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.960·39-s + 0.624·41-s − 0.609·43-s + 9/7·49-s − 0.280·51-s + 1.09·53-s − 0.520·59-s + 0.256·61-s + 0.503·63-s − 1.22·67-s + 0.120·69-s + 1.89·71-s − 1.17·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.784365378\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.784365378\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−16.22288053345283, −15.70183943072049, −15.13886989487546, −14.76735555420315, −13.98612776371657, −13.53829869921177, −13.29157530353897, −12.28436408552535, −11.73244134149859, −11.02754652663925, −10.82879487592251, −9.999546095945803, −9.202206440017978, −8.645802942417153, −8.099081378043382, −7.795411971421114, −6.902146278161043, −6.144616394217345, −5.477677293788279, −4.677950335738700, −4.106893821240556, −3.401916784328080, −2.392510008472681, −1.715790699024902, −0.9181399636161003,
0.9181399636161003, 1.715790699024902, 2.392510008472681, 3.401916784328080, 4.106893821240556, 4.677950335738700, 5.477677293788279, 6.144616394217345, 6.902146278161043, 7.795411971421114, 8.099081378043382, 8.645802942417153, 9.202206440017978, 9.999546095945803, 10.82879487592251, 11.02754652663925, 11.73244134149859, 12.28436408552535, 13.29157530353897, 13.53829869921177, 13.98612776371657, 14.76735555420315, 15.13886989487546, 15.70183943072049, 16.22288053345283