| L(s) = 1 | + 2·2-s + 4·4-s − 11·5-s − 7·7-s + 8·8-s − 22·10-s − 11·11-s − 37·13-s − 14·14-s + 16·16-s + 46·17-s + 15·19-s − 44·20-s − 22·22-s + 92·23-s − 4·25-s − 74·26-s − 28·28-s − 205·29-s + 142·31-s + 32·32-s + 92·34-s + 77·35-s − 431·37-s + 30·38-s − 88·40-s + 8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.983·5-s − 0.377·7-s + 0.353·8-s − 0.695·10-s − 0.301·11-s − 0.789·13-s − 0.267·14-s + 1/4·16-s + 0.656·17-s + 0.181·19-s − 0.491·20-s − 0.213·22-s + 0.834·23-s − 0.0319·25-s − 0.558·26-s − 0.188·28-s − 1.31·29-s + 0.822·31-s + 0.176·32-s + 0.464·34-s + 0.371·35-s − 1.91·37-s + 0.128·38-s − 0.347·40-s + 0.0304·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.195326239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.195326239\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 + p T \) |
| good | 5 | \( 1 + 11 T + p^{3} T^{2} \) |
| 13 | \( 1 + 37 T + p^{3} T^{2} \) |
| 17 | \( 1 - 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - 15 T + p^{3} T^{2} \) |
| 23 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 29 | \( 1 + 205 T + p^{3} T^{2} \) |
| 31 | \( 1 - 142 T + p^{3} T^{2} \) |
| 37 | \( 1 + 431 T + p^{3} T^{2} \) |
| 41 | \( 1 - 8 T + p^{3} T^{2} \) |
| 43 | \( 1 - 448 T + p^{3} T^{2} \) |
| 47 | \( 1 + 149 T + p^{3} T^{2} \) |
| 53 | \( 1 - 672 T + p^{3} T^{2} \) |
| 59 | \( 1 - 615 T + p^{3} T^{2} \) |
| 61 | \( 1 - 322 T + p^{3} T^{2} \) |
| 67 | \( 1 + 411 T + p^{3} T^{2} \) |
| 71 | \( 1 - 968 T + p^{3} T^{2} \) |
| 73 | \( 1 + 227 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 - 1302 T + p^{3} T^{2} \) |
| 89 | \( 1 - 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1736 T + p^{3} T^{2} \) |
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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
−9.220847222239707594739013362673, −8.200879838656664034860145682502, −7.40615572145305758778125232268, −6.91463385880475455540091885890, −5.68702907288142609232285996790, −5.01195995060107833732359118863, −3.96775676310562607389635851391, −3.30435559000835919012181384178, −2.24042998826350543460623148611, −0.64240469912793150772161887219,
0.64240469912793150772161887219, 2.24042998826350543460623148611, 3.30435559000835919012181384178, 3.96775676310562607389635851391, 5.01195995060107833732359118863, 5.68702907288142609232285996790, 6.91463385880475455540091885890, 7.40615572145305758778125232268, 8.200879838656664034860145682502, 9.220847222239707594739013362673
