| L(s) = 1 | + 0.787·2-s + 2.31·3-s − 1.38·4-s + 1.82·6-s + 1.65·7-s − 2.66·8-s + 2.37·9-s + 2.97·11-s − 3.19·12-s + 5.75·13-s + 1.30·14-s + 0.666·16-s − 2.91·17-s + 1.86·18-s + 3.11·19-s + 3.82·21-s + 2.33·22-s − 4.39·23-s − 6.16·24-s + 4.52·26-s − 1.45·27-s − 2.28·28-s + 6.55·29-s − 7.44·31-s + 5.84·32-s + 6.88·33-s − 2.29·34-s + ⋯ |
| L(s) = 1 | + 0.556·2-s + 1.33·3-s − 0.690·4-s + 0.744·6-s + 0.624·7-s − 0.940·8-s + 0.791·9-s + 0.895·11-s − 0.923·12-s + 1.59·13-s + 0.347·14-s + 0.166·16-s − 0.706·17-s + 0.440·18-s + 0.713·19-s + 0.835·21-s + 0.498·22-s − 0.915·23-s − 1.25·24-s + 0.888·26-s − 0.279·27-s − 0.430·28-s + 1.21·29-s − 1.33·31-s + 1.03·32-s + 1.19·33-s − 0.393·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.110489613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.110489613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 0.787T + 2T^{2} \) |
| 3 | \( 1 - 2.31T + 3T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 13 | \( 1 - 5.75T + 13T^{2} \) |
| 17 | \( 1 + 2.91T + 17T^{2} \) |
| 19 | \( 1 - 3.11T + 19T^{2} \) |
| 23 | \( 1 + 4.39T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 - 0.976T + 37T^{2} \) |
| 41 | \( 1 - 9.85T + 41T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 - 6.57T + 53T^{2} \) |
| 59 | \( 1 - 3.45T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 4.71T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 0.162T + 79T^{2} \) |
| 83 | \( 1 - 0.137T + 83T^{2} \) |
| 89 | \( 1 + 8.65T + 89T^{2} \) |
| 97 | \( 1 + 6.52T + 97T^{2} \) |
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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
−9.470297145211357152620176809274, −8.976598777852650800136361275836, −8.425456981036140479180590910224, −7.66677512814430115099355507195, −6.39534573607819558676752003260, −5.53134395473308312515804190684, −4.20068942689790371110325072870, −3.85045218993670853355299672098, −2.77316788773582058949234222673, −1.38789208581493393306461877148,
1.38789208581493393306461877148, 2.77316788773582058949234222673, 3.85045218993670853355299672098, 4.20068942689790371110325072870, 5.53134395473308312515804190684, 6.39534573607819558676752003260, 7.66677512814430115099355507195, 8.425456981036140479180590910224, 8.976598777852650800136361275836, 9.470297145211357152620176809274
