| L(s) = 1 | − 1.08·3-s + 1.08·5-s − 7-s − 1.82·9-s − 5.22·11-s + 6.30·13-s − 1.17·15-s + 3.65·17-s + 4.14·19-s + 1.08·21-s − 1.17·23-s − 3.82·25-s + 5.22·27-s − 8.28·29-s − 5.65·31-s + 5.65·33-s − 1.08·35-s + 2.16·37-s − 6.82·39-s − 7.65·41-s + 5.22·43-s − 1.97·45-s − 8·47-s + 49-s − 3.95·51-s − 4.32·53-s − 5.65·55-s + ⋯ |
| L(s) = 1 | − 0.624·3-s + 0.484·5-s − 0.377·7-s − 0.609·9-s − 1.57·11-s + 1.74·13-s − 0.302·15-s + 0.886·17-s + 0.950·19-s + 0.236·21-s − 0.244·23-s − 0.765·25-s + 1.00·27-s − 1.53·29-s − 1.01·31-s + 0.984·33-s − 0.182·35-s + 0.355·37-s − 1.09·39-s − 1.19·41-s + 0.796·43-s − 0.295·45-s − 1.16·47-s + 0.142·49-s − 0.554·51-s − 0.594·53-s − 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 1.08T + 3T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 11 | \( 1 + 5.22T + 11T^{2} \) |
| 13 | \( 1 - 6.30T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 4.14T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 + 8.28T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 2.16T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 - 5.22T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 4.32T + 53T^{2} \) |
| 59 | \( 1 + 6.30T + 59T^{2} \) |
| 61 | \( 1 - 7.20T + 61T^{2} \) |
| 67 | \( 1 + 7.39T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 0.343T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 5.41T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.912985482583383034091131934673, −8.051547857112211219819547780130, −7.37205830954068663439740696324, −6.10806874249039755051557367326, −5.73509068248932338239097733842, −5.15030013912565917060201334438, −3.68457316220561765646885140016, −2.92941396839128490350689486547, −1.55268222142430473184260242933, 0,
1.55268222142430473184260242933, 2.92941396839128490350689486547, 3.68457316220561765646885140016, 5.15030013912565917060201334438, 5.73509068248932338239097733842, 6.10806874249039755051557367326, 7.37205830954068663439740696324, 8.051547857112211219819547780130, 8.912985482583383034091131934673
