L(s) = 1 | − 3-s + 9-s + 11-s − 2·13-s − 4·17-s + 2·19-s − 5·25-s − 27-s + 4·29-s + 2·31-s − 33-s + 37-s + 2·39-s + 10·41-s − 2·43-s + 8·47-s − 7·49-s + 4·51-s − 10·53-s − 2·57-s + 10·61-s − 8·67-s − 16·71-s + 10·73-s + 5·75-s + 14·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.970·17-s + 0.458·19-s − 25-s − 0.192·27-s + 0.742·29-s + 0.359·31-s − 0.174·33-s + 0.164·37-s + 0.320·39-s + 1.56·41-s − 0.304·43-s + 1.16·47-s − 49-s + 0.560·51-s − 1.37·53-s − 0.264·57-s + 1.28·61-s − 0.977·67-s − 1.89·71-s + 1.17·73-s + 0.577·75-s + 1.57·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.357120505\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.357120505\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.88767787224745, −13.61691825357374, −12.92313557791519, −12.51145907899601, −11.97965521147504, −11.54077884410040, −11.11209401865844, −10.55315420062466, −10.03746032501174, −9.465571805063874, −9.137992200505359, −8.423393701808945, −7.743866678643362, −7.468100573876892, −6.556037013720416, −6.457870681931163, −5.685773686117865, −5.180347721862003, −4.464719760924573, −4.196365731705532, −3.353712394635251, −2.626459077416593, −2.024031202456414, −1.214839515420685, −0.4193531298783204,
0.4193531298783204, 1.214839515420685, 2.024031202456414, 2.626459077416593, 3.353712394635251, 4.196365731705532, 4.464719760924573, 5.180347721862003, 5.685773686117865, 6.457870681931163, 6.556037013720416, 7.468100573876892, 7.743866678643362, 8.423393701808945, 9.137992200505359, 9.465571805063874, 10.03746032501174, 10.55315420062466, 11.11209401865844, 11.54077884410040, 11.97965521147504, 12.51145907899601, 12.92313557791519, 13.61691825357374, 13.88767787224745