| L(s) = 1 | − 3-s + 2·5-s + 9-s − 11-s − 2·15-s − 6·17-s − 4·19-s + 8·23-s − 25-s − 27-s − 10·29-s + 33-s − 6·37-s − 10·41-s − 4·43-s + 2·45-s + 8·47-s − 7·49-s + 6·51-s − 10·53-s − 2·55-s + 4·57-s − 12·59-s + 14·61-s − 12·67-s − 8·69-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s − 0.516·15-s − 1.45·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.174·33-s − 0.986·37-s − 1.56·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s − 49-s + 0.840·51-s − 1.37·53-s − 0.269·55-s + 0.529·57-s − 1.56·59-s + 1.79·61-s − 1.46·67-s − 0.963·69-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5818931189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5818931189\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.53695398479304, −13.35394608836114, −13.08907801164726, −12.45798369792835, −11.90958825346540, −11.25461482361808, −10.80599888543322, −10.66131547722760, −9.867470520183585, −9.404889882478616, −8.954417508640297, −8.491237540489116, −7.746620622161506, −7.111032578214128, −6.626856451258372, −6.298138141343974, −5.568537292058279, −5.137094615495534, −4.687301532394853, −3.965542749130842, −3.295726318134766, −2.526837652647109, −1.850448216774613, −1.494539212693783, −0.2378247516487128,
0.2378247516487128, 1.494539212693783, 1.850448216774613, 2.526837652647109, 3.295726318134766, 3.965542749130842, 4.687301532394853, 5.137094615495534, 5.568537292058279, 6.298138141343974, 6.626856451258372, 7.111032578214128, 7.746620622161506, 8.491237540489116, 8.954417508640297, 9.404889882478616, 9.867470520183585, 10.66131547722760, 10.80599888543322, 11.25461482361808, 11.90958825346540, 12.45798369792835, 13.08907801164726, 13.35394608836114, 13.53695398479304
