| L(s) = 1 | − 3-s − 0.246·5-s − 1.75·7-s + 9-s − 5.65·11-s + 0.246·15-s − 3.80·17-s − 5.58·19-s + 1.75·21-s − 8.34·23-s − 4.93·25-s − 27-s − 5.93·29-s − 5.26·31-s + 5.65·33-s + 0.432·35-s + 3.19·37-s + 0.445·41-s − 1.71·43-s − 0.246·45-s + 6.73·47-s − 3.92·49-s + 3.80·51-s − 1.06·53-s + 1.39·55-s + 5.58·57-s + 13.7·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.110·5-s − 0.662·7-s + 0.333·9-s − 1.70·11-s + 0.0637·15-s − 0.922·17-s − 1.28·19-s + 0.382·21-s − 1.74·23-s − 0.987·25-s − 0.192·27-s − 1.10·29-s − 0.946·31-s + 0.984·33-s + 0.0731·35-s + 0.525·37-s + 0.0695·41-s − 0.261·43-s − 0.0368·45-s + 0.982·47-s − 0.560·49-s + 0.532·51-s − 0.145·53-s + 0.188·55-s + 0.739·57-s + 1.78·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09477376016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09477376016\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.246T + 5T^{2} \) |
| 7 | \( 1 + 1.75T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 + 8.34T + 23T^{2} \) |
| 29 | \( 1 + 5.93T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 - 0.445T + 41T^{2} \) |
| 43 | \( 1 + 1.71T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 + 1.06T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 8.51T + 61T^{2} \) |
| 67 | \( 1 + 5.96T + 67T^{2} \) |
| 71 | \( 1 - 5.71T + 71T^{2} \) |
| 73 | \( 1 - 7.35T + 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 0.137T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−7.80516015815629640094902300356, −7.14405151654494129097854825680, −6.27961834629204047044283701557, −5.83720608021905658061622855080, −5.13752811212204484557031219713, −4.23414292691228230980107177706, −3.68702181148244587722155528740, −2.46782766306413937563028598130, −1.96664725469812917344432643134, −0.14380809828986967109213364828,
0.14380809828986967109213364828, 1.96664725469812917344432643134, 2.46782766306413937563028598130, 3.68702181148244587722155528740, 4.23414292691228230980107177706, 5.13752811212204484557031219713, 5.83720608021905658061622855080, 6.27961834629204047044283701557, 7.14405151654494129097854825680, 7.80516015815629640094902300356
