| L(s) = 1 | + 1.66·3-s + 4.35·7-s − 0.242·9-s − 0.760·11-s − 3.53·13-s − 17-s − 0.972·19-s + 7.23·21-s − 7.47·23-s − 5.38·27-s − 5.25·29-s − 8.62·31-s − 1.26·33-s + 5.94·37-s − 5.87·39-s − 4.29·41-s − 3.98·43-s − 6.28·47-s + 11.9·49-s − 1.66·51-s − 1.54·53-s − 1.61·57-s − 2.66·59-s − 3.32·61-s − 1.05·63-s − 15.9·67-s − 12.4·69-s + ⋯ |
| L(s) = 1 | + 0.958·3-s + 1.64·7-s − 0.0807·9-s − 0.229·11-s − 0.980·13-s − 0.242·17-s − 0.223·19-s + 1.57·21-s − 1.55·23-s − 1.03·27-s − 0.976·29-s − 1.54·31-s − 0.219·33-s + 0.976·37-s − 0.940·39-s − 0.670·41-s − 0.608·43-s − 0.916·47-s + 1.71·49-s − 0.232·51-s − 0.211·53-s − 0.213·57-s − 0.347·59-s − 0.425·61-s − 0.132·63-s − 1.95·67-s − 1.49·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.66T + 3T^{2} \) |
| 7 | \( 1 - 4.35T + 7T^{2} \) |
| 11 | \( 1 + 0.760T + 11T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 19 | \( 1 + 0.972T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 + 5.25T + 29T^{2} \) |
| 31 | \( 1 + 8.62T + 31T^{2} \) |
| 37 | \( 1 - 5.94T + 37T^{2} \) |
| 41 | \( 1 + 4.29T + 41T^{2} \) |
| 43 | \( 1 + 3.98T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 + 1.54T + 53T^{2} \) |
| 59 | \( 1 + 2.66T + 59T^{2} \) |
| 61 | \( 1 + 3.32T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 4.45T + 79T^{2} \) |
| 83 | \( 1 - 6.71T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 7.19T + 97T^{2} \) |
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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
−7.73842187682324251354326876693, −7.34188438248811962580268943913, −6.13714357662152721726414200057, −5.38600531616964311213613658719, −4.70793486613998069053535787237, −3.99661094688917342918179890359, −3.10266189628977753031884735505, −2.03640187038463238372071018291, −1.82917394292822466015533169363, 0,
1.82917394292822466015533169363, 2.03640187038463238372071018291, 3.10266189628977753031884735505, 3.99661094688917342918179890359, 4.70793486613998069053535787237, 5.38600531616964311213613658719, 6.13714357662152721726414200057, 7.34188438248811962580268943913, 7.73842187682324251354326876693
