| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s − 13-s + 15-s − 17-s + 2·19-s − 2·21-s + 25-s + 27-s + 6·29-s + 4·31-s − 2·35-s + 10·37-s − 39-s − 6·41-s − 10·43-s + 45-s − 3·49-s − 51-s + 6·53-s + 2·57-s − 6·59-s − 14·61-s − 2·63-s − 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.458·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.338·35-s + 1.64·37-s − 0.160·39-s − 0.937·41-s − 1.52·43-s + 0.149·45-s − 3/7·49-s − 0.140·51-s + 0.824·53-s + 0.264·57-s − 0.781·59-s − 1.79·61-s − 0.251·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.045055609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.045055609\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.20756319845058, −12.58674458636042, −12.01294116282588, −11.87083744955640, −10.97461479344088, −10.66639002559104, −9.982681303546423, −9.691379679789250, −9.412589055070616, −8.788571885390941, −8.198351560088098, −7.956249157207384, −7.245471088573706, −6.649049983366146, −6.426265121546084, −5.891960643602001, −4.997880316247903, −4.876285565771924, −4.072669653694584, −3.508900897774582, −2.928152518127058, −2.597137793186307, −1.887255675438033, −1.205843444075903, −0.4821740485309796,
0.4821740485309796, 1.205843444075903, 1.887255675438033, 2.597137793186307, 2.928152518127058, 3.508900897774582, 4.072669653694584, 4.876285565771924, 4.997880316247903, 5.891960643602001, 6.426265121546084, 6.649049983366146, 7.245471088573706, 7.956249157207384, 8.198351560088098, 8.788571885390941, 9.412589055070616, 9.691379679789250, 9.982681303546423, 10.66639002559104, 10.97461479344088, 11.87083744955640, 12.01294116282588, 12.58674458636042, 13.20756319845058
