L(s) = 1 | + 3·3-s − 7-s + 6·9-s + 2·11-s − 13-s + 3·17-s − 19-s − 3·21-s + 3·23-s − 5·25-s + 9·27-s + 3·29-s + 8·31-s + 6·33-s − 10·37-s − 3·39-s − 12·41-s + 8·43-s − 8·47-s − 6·49-s + 9·51-s − 9·53-s − 3·57-s − 5·59-s + 10·61-s − 6·63-s + 7·67-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.377·7-s + 2·9-s + 0.603·11-s − 0.277·13-s + 0.727·17-s − 0.229·19-s − 0.654·21-s + 0.625·23-s − 25-s + 1.73·27-s + 0.557·29-s + 1.43·31-s + 1.04·33-s − 1.64·37-s − 0.480·39-s − 1.87·41-s + 1.21·43-s − 1.16·47-s − 6/7·49-s + 1.26·51-s − 1.23·53-s − 0.397·57-s − 0.650·59-s + 1.28·61-s − 0.755·63-s + 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.587839377\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.587839377\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−10.15555477800038159845119846559, −9.752900189125497232452224011820, −8.796590556881761178241246784034, −8.197820813644998743406881037051, −7.28048593381483137900855724231, −6.39357456655427571414069968224, −4.82665740673161335065260299640, −3.66834687648261549407687934911, −2.94857221124987493994611399801, −1.66549496948467502043296521631,
1.66549496948467502043296521631, 2.94857221124987493994611399801, 3.66834687648261549407687934911, 4.82665740673161335065260299640, 6.39357456655427571414069968224, 7.28048593381483137900855724231, 8.197820813644998743406881037051, 8.796590556881761178241246784034, 9.752900189125497232452224011820, 10.15555477800038159845119846559