| L(s) = 1 | + 3-s + 2·5-s + 7-s − 2·9-s − 11-s + 6·13-s + 2·15-s − 4·17-s + 8·19-s + 21-s + 6·23-s − 25-s − 5·27-s − 2·29-s − 4·31-s − 33-s + 2·35-s + 37-s + 6·39-s + 7·41-s − 2·43-s − 4·45-s + 9·47-s − 6·49-s − 4·51-s + 3·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s − 2/3·9-s − 0.301·11-s + 1.66·13-s + 0.516·15-s − 0.970·17-s + 1.83·19-s + 0.218·21-s + 1.25·23-s − 1/5·25-s − 0.962·27-s − 0.371·29-s − 0.718·31-s − 0.174·33-s + 0.338·35-s + 0.164·37-s + 0.960·39-s + 1.09·41-s − 0.304·43-s − 0.596·45-s + 1.31·47-s − 6/7·49-s − 0.560·51-s + 0.412·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.808314032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.808314032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.106986862927494823398773273392, −8.323706156991376435426293982790, −7.56200991672588442793948035593, −6.62515548141648090625940324832, −5.68536278134072713474064800006, −5.29977652365226555112580419353, −3.96828612768597928011477043556, −3.11027536009172942087782204014, −2.21589821767858410532420583902, −1.13230615835739562152636563080,
1.13230615835739562152636563080, 2.21589821767858410532420583902, 3.11027536009172942087782204014, 3.96828612768597928011477043556, 5.29977652365226555112580419353, 5.68536278134072713474064800006, 6.62515548141648090625940324832, 7.56200991672588442793948035593, 8.323706156991376435426293982790, 9.106986862927494823398773273392
