| L(s) = 1 | − 3.12·3-s + 2·5-s + 0.516·7-s + 6.77·9-s − 0.538·11-s − 0.295·13-s − 6.25·15-s − 6.99·17-s + 4.06·19-s − 1.61·21-s − 0.781·23-s − 25-s − 11.7·27-s + 0.811·29-s − 3.07·31-s + 1.68·33-s + 1.03·35-s − 3.95·37-s + 0.923·39-s + 0.743·41-s − 0.590·43-s + 13.5·45-s + 3.47·47-s − 6.73·49-s + 21.8·51-s + 6.69·53-s − 1.07·55-s + ⋯ |
| L(s) = 1 | − 1.80·3-s + 0.894·5-s + 0.195·7-s + 2.25·9-s − 0.162·11-s − 0.0819·13-s − 1.61·15-s − 1.69·17-s + 0.932·19-s − 0.352·21-s − 0.162·23-s − 0.200·25-s − 2.26·27-s + 0.150·29-s − 0.552·31-s + 0.292·33-s + 0.174·35-s − 0.650·37-s + 0.147·39-s + 0.116·41-s − 0.0900·43-s + 2.01·45-s + 0.506·47-s − 0.961·49-s + 3.06·51-s + 0.920·53-s − 0.145·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 3.12T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 0.516T + 7T^{2} \) |
| 11 | \( 1 + 0.538T + 11T^{2} \) |
| 13 | \( 1 + 0.295T + 13T^{2} \) |
| 17 | \( 1 + 6.99T + 17T^{2} \) |
| 19 | \( 1 - 4.06T + 19T^{2} \) |
| 23 | \( 1 + 0.781T + 23T^{2} \) |
| 29 | \( 1 - 0.811T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 + 3.95T + 37T^{2} \) |
| 41 | \( 1 - 0.743T + 41T^{2} \) |
| 43 | \( 1 + 0.590T + 43T^{2} \) |
| 47 | \( 1 - 3.47T + 47T^{2} \) |
| 53 | \( 1 - 6.69T + 53T^{2} \) |
| 59 | \( 1 - 8.91T + 59T^{2} \) |
| 61 | \( 1 - 4.63T + 61T^{2} \) |
| 67 | \( 1 + 6.55T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 7.77T + 73T^{2} \) |
| 79 | \( 1 - 4.88T + 79T^{2} \) |
| 83 | \( 1 - 1.11T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−8.554227129519632502645725976266, −7.34173693078020215608619395131, −6.77222800059523481214127563228, −6.04311579998293243020272576888, −5.43129739290500507865950871602, −4.83516891842823912889026686103, −3.94236122400812901502826016898, −2.33586273426026262679737069834, −1.34624105845237039350469801856, 0,
1.34624105845237039350469801856, 2.33586273426026262679737069834, 3.94236122400812901502826016898, 4.83516891842823912889026686103, 5.43129739290500507865950871602, 6.04311579998293243020272576888, 6.77222800059523481214127563228, 7.34173693078020215608619395131, 8.554227129519632502645725976266
