| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4889 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4889 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.570128605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.570128605\) |
| \(L(1)\) |
\(\approx\) |
\(1.894378329\) |
| \(L(1)\) |
\(\approx\) |
\(1.894378329\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4889 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.97336622430891018039514268606, −17.24050985666825745472659587042, −16.6278561046782617404567626880, −16.18239437789150477191189963159, −15.4820872248899634967111102272, −14.75110390663717247222070002653, −13.80283062926511464544609950903, −13.34190671248844103653568604134, −12.85131764416054537352061996790, −12.192927694033190230157089454683, −11.29557944508291930034373564281, −10.966950810580073001159557390639, −10.11627801340058427504638305544, −9.42722646399739371829023756102, −8.76572143772702297136216538804, −7.132392024861336004416371143195, −6.90614764934988028407055115529, −6.22817759050770464600153850313, −5.60378502378485990923543047430, −5.16884773917158136218688787058, −4.0102536806828832447432733126, −3.61933510372583513352305340974, −2.55267618912982362202123742216, −1.63638651749137580405870508754, −0.94634939937389980499707156469,
0.94634939937389980499707156469, 1.63638651749137580405870508754, 2.55267618912982362202123742216, 3.61933510372583513352305340974, 4.0102536806828832447432733126, 5.16884773917158136218688787058, 5.60378502378485990923543047430, 6.22817759050770464600153850313, 6.90614764934988028407055115529, 7.132392024861336004416371143195, 8.76572143772702297136216538804, 9.42722646399739371829023756102, 10.11627801340058427504638305544, 10.966950810580073001159557390639, 11.29557944508291930034373564281, 12.192927694033190230157089454683, 12.85131764416054537352061996790, 13.34190671248844103653568604134, 13.80283062926511464544609950903, 14.75110390663717247222070002653, 15.4820872248899634967111102272, 16.18239437789150477191189963159, 16.6278561046782617404567626880, 17.24050985666825745472659587042, 17.97336622430891018039514268606
