| L(s) = 1 | + 3-s − 3·5-s − 7-s + 9-s − 2·11-s − 13-s − 3·15-s + 2·17-s − 21-s + 4·25-s + 27-s − 9·29-s + 2·31-s − 2·33-s + 3·35-s + 3·37-s − 39-s + 7·41-s + 9·43-s − 3·45-s − 47-s + 49-s + 2·51-s + 6·55-s + 6·61-s − 63-s + 3·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.774·15-s + 0.485·17-s − 0.218·21-s + 4/5·25-s + 0.192·27-s − 1.67·29-s + 0.359·31-s − 0.348·33-s + 0.507·35-s + 0.493·37-s − 0.160·39-s + 1.09·41-s + 1.37·43-s − 0.447·45-s − 0.145·47-s + 1/7·49-s + 0.280·51-s + 0.809·55-s + 0.768·61-s − 0.125·63-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.734560443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.734560443\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−13.04353677734969, −12.80067675080044, −12.21516892374272, −11.85181438581967, −11.28490801689195, −10.81080523382466, −10.46320830181949, −9.618965076808987, −9.473446101369420, −8.859445181607410, −8.228471985897482, −7.870842378652785, −7.449197936501773, −7.194029593665712, −6.418482394560428, −5.804843399358592, −5.312521136876452, −4.534891276376318, −4.220779338352217, −3.477624973916679, −3.341128772962794, −2.470511148132549, −2.086602015468861, −0.9926385473903121, −0.4260757279193961,
0.4260757279193961, 0.9926385473903121, 2.086602015468861, 2.470511148132549, 3.341128772962794, 3.477624973916679, 4.220779338352217, 4.534891276376318, 5.312521136876452, 5.804843399358592, 6.418482394560428, 7.194029593665712, 7.449197936501773, 7.870842378652785, 8.228471985897482, 8.859445181607410, 9.473446101369420, 9.618965076808987, 10.46320830181949, 10.81080523382466, 11.28490801689195, 11.85181438581967, 12.21516892374272, 12.80067675080044, 13.04353677734969
