| L(s) = 1 | − 2-s + 4-s + 4.05·7-s − 8-s + 0.466·11-s + 0.985·13-s − 4.05·14-s + 16-s − 4.94·17-s + 19-s − 0.466·22-s + 2.22·23-s − 0.985·26-s + 4.05·28-s − 2.43·29-s + 0.715·31-s − 32-s + 4.94·34-s − 10.0·37-s − 38-s − 10.1·41-s + 0.466·43-s + 0.466·44-s − 2.22·46-s − 8.79·47-s + 9.45·49-s + 0.985·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.53·7-s − 0.353·8-s + 0.140·11-s + 0.273·13-s − 1.08·14-s + 0.250·16-s − 1.19·17-s + 0.229·19-s − 0.0993·22-s + 0.464·23-s − 0.193·26-s + 0.766·28-s − 0.452·29-s + 0.128·31-s − 0.176·32-s + 0.848·34-s − 1.64·37-s − 0.162·38-s − 1.59·41-s + 0.0710·43-s + 0.0702·44-s − 0.328·46-s − 1.28·47-s + 1.35·49-s + 0.136·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4.05T + 7T^{2} \) |
| 11 | \( 1 - 0.466T + 11T^{2} \) |
| 13 | \( 1 - 0.985T + 13T^{2} \) |
| 17 | \( 1 + 4.94T + 17T^{2} \) |
| 23 | \( 1 - 2.22T + 23T^{2} \) |
| 29 | \( 1 + 2.43T + 29T^{2} \) |
| 31 | \( 1 - 0.715T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 0.466T + 43T^{2} \) |
| 47 | \( 1 + 8.79T + 47T^{2} \) |
| 53 | \( 1 - 3.89T + 53T^{2} \) |
| 59 | \( 1 + 7.42T + 59T^{2} \) |
| 61 | \( 1 + 6.45T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 6.14T + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 - 1.85T + 79T^{2} \) |
| 83 | \( 1 + 4.37T + 83T^{2} \) |
| 89 | \( 1 - 6.02T + 89T^{2} \) |
| 97 | \( 1 - 6.60T + 97T^{2} \) |
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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
−7.54305699379627351594681937279, −6.90536457576777928108686322618, −6.22946946727873391404459852796, −5.22843021856657947207237697903, −4.79156367110954403346036722883, −3.87783638870843789433096367219, −2.91315449046528482256645318391, −1.84106000307685470302927890103, −1.43474474840938311673767147394, 0,
1.43474474840938311673767147394, 1.84106000307685470302927890103, 2.91315449046528482256645318391, 3.87783638870843789433096367219, 4.79156367110954403346036722883, 5.22843021856657947207237697903, 6.22946946727873391404459852796, 6.90536457576777928108686322618, 7.54305699379627351594681937279
