| L(s) = 1 | + 1.75·3-s − 3.24·5-s + 7-s + 0.0892·9-s + 11-s + 13-s − 5.69·15-s + 0.0754·17-s − 6.53·19-s + 1.75·21-s − 8.67·23-s + 5.51·25-s − 5.11·27-s + 3.22·29-s + 5.81·31-s + 1.75·33-s − 3.24·35-s + 1.83·37-s + 1.75·39-s + 9.76·41-s − 3.89·43-s − 0.289·45-s − 1.46·47-s + 49-s + 0.132·51-s + 8.85·53-s − 3.24·55-s + ⋯ |
| L(s) = 1 | + 1.01·3-s − 1.44·5-s + 0.377·7-s + 0.0297·9-s + 0.301·11-s + 0.277·13-s − 1.47·15-s + 0.0183·17-s − 1.49·19-s + 0.383·21-s − 1.80·23-s + 1.10·25-s − 0.984·27-s + 0.599·29-s + 1.04·31-s + 0.305·33-s − 0.548·35-s + 0.301·37-s + 0.281·39-s + 1.52·41-s − 0.593·43-s − 0.0431·45-s − 0.213·47-s + 0.142·49-s + 0.0185·51-s + 1.21·53-s − 0.437·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.794585630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.794585630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.75T + 3T^{2} \) |
| 5 | \( 1 + 3.24T + 5T^{2} \) |
| 17 | \( 1 - 0.0754T + 17T^{2} \) |
| 19 | \( 1 + 6.53T + 19T^{2} \) |
| 23 | \( 1 + 8.67T + 23T^{2} \) |
| 29 | \( 1 - 3.22T + 29T^{2} \) |
| 31 | \( 1 - 5.81T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 - 9.76T + 41T^{2} \) |
| 43 | \( 1 + 3.89T + 43T^{2} \) |
| 47 | \( 1 + 1.46T + 47T^{2} \) |
| 53 | \( 1 - 8.85T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 0.744T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 7.68T + 71T^{2} \) |
| 73 | \( 1 - 9.86T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 - 2.35T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.012483767732378318106812571441, −7.44309371958540296571938136986, −6.53734911677438045238948837214, −5.88123703901113967300717061400, −4.69250921357030565848379292494, −4.03445426963264991454453705307, −3.72019182044754018123814497559, −2.68524410585060574553488049923, −2.00229952290265358042456809243, −0.61013482037413957357490402914,
0.61013482037413957357490402914, 2.00229952290265358042456809243, 2.68524410585060574553488049923, 3.72019182044754018123814497559, 4.03445426963264991454453705307, 4.69250921357030565848379292494, 5.88123703901113967300717061400, 6.53734911677438045238948837214, 7.44309371958540296571938136986, 8.012483767732378318106812571441
