| L(s) = 1 | + 2.31·2-s + 5·3-s − 2.63·4-s + 11.5·6-s − 24.6·8-s − 2·9-s + 46.2·11-s − 13.1·12-s − 61.3·13-s − 36·16-s + 101.·17-s − 4.63·18-s − 3.66·19-s + 107.·22-s − 84.8·23-s − 123.·24-s − 142.·26-s − 145·27-s + 30.1·29-s − 188.·31-s + 113.·32-s + 231.·33-s + 234.·34-s + 5.26·36-s − 18.0·37-s − 8.49·38-s − 306.·39-s + ⋯ |
| L(s) = 1 | + 0.819·2-s + 0.962·3-s − 0.329·4-s + 0.788·6-s − 1.08·8-s − 0.0740·9-s + 1.26·11-s − 0.316·12-s − 1.30·13-s − 0.562·16-s + 1.44·17-s − 0.0606·18-s − 0.0442·19-s + 1.03·22-s − 0.769·23-s − 1.04·24-s − 1.07·26-s − 1.03·27-s + 0.193·29-s − 1.09·31-s + 0.627·32-s + 1.22·33-s + 1.18·34-s + 0.0243·36-s − 0.0802·37-s − 0.0362·38-s − 1.25·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.31T + 8T^{2} \) |
| 3 | \( 1 - 5T + 27T^{2} \) |
| 11 | \( 1 - 46.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 3.66T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 30.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 18.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 481.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 97.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 117.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 667.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 57.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 738.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 552.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 740.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 233.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 683.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 218.T + 9.12e5T^{2} \) |
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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
−9.107499168023484510921021889426, −8.142042474502259274595092274504, −7.39514770475560625027988294770, −6.25918735833701851852579826473, −5.42529955971033518814233429518, −4.48859740726931126982775771467, −3.56991308006902941876339759207, −2.98657849416034593933676402480, −1.70382570504049603897189393871, 0,
1.70382570504049603897189393871, 2.98657849416034593933676402480, 3.56991308006902941876339759207, 4.48859740726931126982775771467, 5.42529955971033518814233429518, 6.25918735833701851852579826473, 7.39514770475560625027988294770, 8.142042474502259274595092274504, 9.107499168023484510921021889426
