| L(s) = 1 | − 2-s − 1.35·3-s + 4-s − 0.395·5-s + 1.35·6-s + 1.50·7-s − 8-s − 1.17·9-s + 0.395·10-s + 1.06·11-s − 1.35·12-s − 0.679·13-s − 1.50·14-s + 0.535·15-s + 16-s − 0.116·17-s + 1.17·18-s + 1.90·19-s − 0.395·20-s − 2.03·21-s − 1.06·22-s − 2.44·23-s + 1.35·24-s − 4.84·25-s + 0.679·26-s + 5.64·27-s + 1.50·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.780·3-s + 0.5·4-s − 0.176·5-s + 0.552·6-s + 0.569·7-s − 0.353·8-s − 0.390·9-s + 0.125·10-s + 0.322·11-s − 0.390·12-s − 0.188·13-s − 0.402·14-s + 0.138·15-s + 0.250·16-s − 0.0282·17-s + 0.275·18-s + 0.436·19-s − 0.0884·20-s − 0.444·21-s − 0.227·22-s − 0.510·23-s + 0.276·24-s − 0.968·25-s + 0.133·26-s + 1.08·27-s + 0.284·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 5 | \( 1 + 0.395T + 5T^{2} \) |
| 7 | \( 1 - 1.50T + 7T^{2} \) |
| 11 | \( 1 - 1.06T + 11T^{2} \) |
| 13 | \( 1 + 0.679T + 13T^{2} \) |
| 17 | \( 1 + 0.116T + 17T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 - 0.146T + 41T^{2} \) |
| 43 | \( 1 - 2.52T + 43T^{2} \) |
| 47 | \( 1 - 8.25T + 47T^{2} \) |
| 53 | \( 1 + 1.27T + 53T^{2} \) |
| 59 | \( 1 + 6.85T + 59T^{2} \) |
| 61 | \( 1 - 0.454T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 2.75T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 5.48T + 89T^{2} \) |
| 97 | \( 1 + 4.05T + 97T^{2} \) |
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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
−8.069091618981312110820636794038, −7.47339612801626565105238774089, −6.58488485666629240156297834261, −5.95379905964147666736557199128, −5.21936330560177096793688822999, −4.39054463228402938476711059000, −3.32186946357090130259501932755, −2.25316826270050630005655851671, −1.17450932037387402687802262109, 0,
1.17450932037387402687802262109, 2.25316826270050630005655851671, 3.32186946357090130259501932755, 4.39054463228402938476711059000, 5.21936330560177096793688822999, 5.95379905964147666736557199128, 6.58488485666629240156297834261, 7.47339612801626565105238774089, 8.069091618981312110820636794038
