| L(s) = 1 | + 2-s + 0.996·3-s + 4-s − 0.445·5-s + 0.996·6-s + 2.40·7-s + 8-s − 2.00·9-s − 0.445·10-s − 0.565·11-s + 0.996·12-s − 4.92·13-s + 2.40·14-s − 0.443·15-s + 16-s − 7.95·17-s − 2.00·18-s + 2.57·19-s − 0.445·20-s + 2.39·21-s − 0.565·22-s − 3.82·23-s + 0.996·24-s − 4.80·25-s − 4.92·26-s − 4.98·27-s + 2.40·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.575·3-s + 0.5·4-s − 0.199·5-s + 0.406·6-s + 0.907·7-s + 0.353·8-s − 0.669·9-s − 0.140·10-s − 0.170·11-s + 0.287·12-s − 1.36·13-s + 0.642·14-s − 0.114·15-s + 0.250·16-s − 1.92·17-s − 0.473·18-s + 0.591·19-s − 0.0995·20-s + 0.522·21-s − 0.120·22-s − 0.798·23-s + 0.203·24-s − 0.960·25-s − 0.966·26-s − 0.960·27-s + 0.453·28-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 | \( 1 - 0.996T + 3T^{2} \) |
| 5 | \( 1 + 0.445T + 5T^{2} \) |
| 7 | \( 1 - 2.40T + 7T^{2} \) |
| 11 | \( 1 + 0.565T + 11T^{2} \) |
| 13 | \( 1 + 4.92T + 13T^{2} \) |
| 17 | \( 1 + 7.95T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 + 3.82T + 23T^{2} \) |
| 29 | \( 1 + 7.72T + 29T^{2} \) |
| 31 | \( 1 + 0.131T + 31T^{2} \) |
| 37 | \( 1 - 1.54T + 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 + 6.39T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 - 7.19T + 53T^{2} \) |
| 59 | \( 1 - 7.86T + 59T^{2} \) |
| 61 | \( 1 - 8.74T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 3.95T + 71T^{2} \) |
| 73 | \( 1 - 9.73T + 73T^{2} \) |
| 79 | \( 1 + 9.57T + 79T^{2} \) |
| 83 | \( 1 - 8.54T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 + 9.73T + 97T^{2} \) |
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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
−7.925852708711479833513061356179, −7.48266605133896867970785539909, −6.62037179559170509330591449150, −5.64674465672688481145726034630, −5.01654161478193927201703377313, −4.27316544651126833189602703679, −3.49052591838730990431558225886, −2.32613867840662128776893010313, −2.03661910453280826411629093858, 0,
2.03661910453280826411629093858, 2.32613867840662128776893010313, 3.49052591838730990431558225886, 4.27316544651126833189602703679, 5.01654161478193927201703377313, 5.64674465672688481145726034630, 6.62037179559170509330591449150, 7.48266605133896867970785539909, 7.925852708711479833513061356179
