| L(s) = 1 | + 0.631·3-s + 2.09·7-s − 2.60·9-s + 5.36·11-s − 5.07·13-s − 5.87·17-s + 6.18·19-s + 1.32·21-s − 3.55·23-s − 3.53·27-s + 0.899·29-s + 10.0·31-s + 3.39·33-s − 3.41·37-s − 3.20·39-s + 5.42·41-s + 2.70·43-s − 1.71·47-s − 2.61·49-s − 3.71·51-s + 8.89·53-s + 3.91·57-s − 4.46·59-s − 4.93·61-s − 5.44·63-s − 3.83·67-s − 2.24·69-s + ⋯ |
| L(s) = 1 | + 0.364·3-s + 0.791·7-s − 0.866·9-s + 1.61·11-s − 1.40·13-s − 1.42·17-s + 1.41·19-s + 0.288·21-s − 0.740·23-s − 0.681·27-s + 0.167·29-s + 1.79·31-s + 0.590·33-s − 0.561·37-s − 0.513·39-s + 0.847·41-s + 0.411·43-s − 0.249·47-s − 0.373·49-s − 0.519·51-s + 1.22·53-s + 0.517·57-s − 0.581·59-s − 0.632·61-s − 0.686·63-s − 0.467·67-s − 0.270·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.421945736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.421945736\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.631T + 3T^{2} \) |
| 7 | \( 1 - 2.09T + 7T^{2} \) |
| 11 | \( 1 - 5.36T + 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 - 6.18T + 19T^{2} \) |
| 23 | \( 1 + 3.55T + 23T^{2} \) |
| 29 | \( 1 - 0.899T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 - 5.42T + 41T^{2} \) |
| 43 | \( 1 - 2.70T + 43T^{2} \) |
| 47 | \( 1 + 1.71T + 47T^{2} \) |
| 53 | \( 1 - 8.89T + 53T^{2} \) |
| 59 | \( 1 + 4.46T + 59T^{2} \) |
| 61 | \( 1 + 4.93T + 61T^{2} \) |
| 67 | \( 1 + 3.83T + 67T^{2} \) |
| 71 | \( 1 - 4.09T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 1.55T + 79T^{2} \) |
| 83 | \( 1 + 1.21T + 83T^{2} \) |
| 89 | \( 1 - 7.01T + 89T^{2} \) |
| 97 | \( 1 + 3.47T + 97T^{2} \) |
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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
−7.73711098728792508759095309981, −6.98645801783839163520824961765, −6.38011757294539922664197175229, −5.60083725999712588626059348430, −4.75217986249406732689128800643, −4.31036895471745338174620177886, −3.35699794006199502129083671183, −2.53039299239388134084178142179, −1.84395352354450381744106917490, −0.72081518097009149509125777026,
0.72081518097009149509125777026, 1.84395352354450381744106917490, 2.53039299239388134084178142179, 3.35699794006199502129083671183, 4.31036895471745338174620177886, 4.75217986249406732689128800643, 5.60083725999712588626059348430, 6.38011757294539922664197175229, 6.98645801783839163520824961765, 7.73711098728792508759095309981
