| L(s) = 1 | − 7-s − 3·9-s + 4·11-s + 4·17-s − 2·19-s + 4·23-s − 5·25-s − 10·29-s + 6·31-s − 6·41-s − 4·43-s − 12·47-s + 49-s − 6·53-s + 2·59-s − 12·61-s + 3·63-s − 12·67-s − 8·71-s + 2·73-s − 4·77-s + 12·79-s + 9·81-s − 16·83-s − 12·89-s + 8·97-s − 12·99-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s + 1.20·11-s + 0.970·17-s − 0.458·19-s + 0.834·23-s − 25-s − 1.85·29-s + 1.07·31-s − 0.937·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.260·59-s − 1.53·61-s + 0.377·63-s − 1.46·67-s − 0.949·71-s + 0.234·73-s − 0.455·77-s + 1.35·79-s + 81-s − 1.75·83-s − 1.27·89-s + 0.812·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.146755953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.146755953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.06075306122454, −13.47173445407280, −13.23126957435153, −12.37050038140323, −12.10076604083860, −11.42556577682403, −11.29671856960058, −10.55556822845576, −9.840007042968128, −9.573707494927366, −8.995061095191942, −8.486200491201077, −7.991709994803612, −7.355278054366115, −6.806285911772331, −6.094668343847271, −5.950389988252128, −5.173807873974492, −4.574728769659888, −3.847034825140431, −3.263010394734153, −2.953265068011629, −1.857750890687500, −1.433540710048625, −0.3459746332200766,
0.3459746332200766, 1.433540710048625, 1.857750890687500, 2.953265068011629, 3.263010394734153, 3.847034825140431, 4.574728769659888, 5.173807873974492, 5.950389988252128, 6.094668343847271, 6.806285911772331, 7.355278054366115, 7.991709994803612, 8.486200491201077, 8.995061095191942, 9.573707494927366, 9.840007042968128, 10.55556822845576, 11.29671856960058, 11.42556577682403, 12.10076604083860, 12.37050038140323, 13.23126957435153, 13.47173445407280, 14.06075306122454
