| L(s) = 1 | − 0.801·3-s − 1.60·7-s − 2.35·9-s − 6.09·11-s − 1.35·13-s − 1.60·17-s + 1.78·19-s + 1.28·21-s − 23-s + 4.29·27-s − 4.82·29-s + 6.15·31-s + 4.89·33-s − 9.87·37-s + 1.08·39-s − 8.89·41-s − 11.3·43-s + 9.85·47-s − 4.42·49-s + 1.28·51-s + 9.20·53-s − 1.42·57-s − 7.27·59-s + 0.933·61-s + 3.78·63-s + 9.42·67-s + 0.801·69-s + ⋯ |
| L(s) = 1 | − 0.462·3-s − 0.606·7-s − 0.785·9-s − 1.83·11-s − 0.376·13-s − 0.388·17-s + 0.408·19-s + 0.280·21-s − 0.208·23-s + 0.826·27-s − 0.896·29-s + 1.10·31-s + 0.851·33-s − 1.62·37-s + 0.174·39-s − 1.38·41-s − 1.73·43-s + 1.43·47-s − 0.632·49-s + 0.180·51-s + 1.26·53-s − 0.189·57-s − 0.947·59-s + 0.119·61-s + 0.476·63-s + 1.15·67-s + 0.0965·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5000239931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5000239931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.801T + 3T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 + 6.09T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 - 6.15T + 31T^{2} \) |
| 37 | \( 1 + 9.87T + 37T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 9.85T + 47T^{2} \) |
| 53 | \( 1 - 9.20T + 53T^{2} \) |
| 59 | \( 1 + 7.27T + 59T^{2} \) |
| 61 | \( 1 - 0.933T + 61T^{2} \) |
| 67 | \( 1 - 9.42T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 2.60T + 73T^{2} \) |
| 79 | \( 1 + 5.16T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.336766510640660917209872405795, −7.58780996156104476696554756543, −6.82410777421114822804931027757, −6.09977662577905792800304337577, −5.16508678551875178399171338200, −5.05831940585531226117667534419, −3.61740725387602433086897239574, −2.89399246098083307640500674180, −2.08547375876196695026451774694, −0.37328064475887228071451918765,
0.37328064475887228071451918765, 2.08547375876196695026451774694, 2.89399246098083307640500674180, 3.61740725387602433086897239574, 5.05831940585531226117667534419, 5.16508678551875178399171338200, 6.09977662577905792800304337577, 6.82410777421114822804931027757, 7.58780996156104476696554756543, 8.336766510640660917209872405795
