| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 4·13-s + 15-s + 2·19-s − 21-s − 23-s + 25-s + 27-s − 6·29-s − 2·31-s − 35-s + 10·37-s + 4·39-s + 6·41-s − 4·43-s + 45-s − 6·47-s + 49-s + 6·53-s + 2·57-s + 12·59-s + 10·61-s − 63-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.10·13-s + 0.258·15-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.169·35-s + 1.64·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s − 0.875·47-s + 1/7·49-s + 0.824·53-s + 0.264·57-s + 1.56·59-s + 1.28·61-s − 0.125·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.270040218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.270040218\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.35956634665767, −12.87001334914578, −12.67883749329966, −11.79307913860414, −11.29557384045620, −11.09641979959068, −10.21284858614106, −9.998754573759118, −9.420484116361194, −9.021712600168147, −8.555788148368428, −7.981249882829842, −7.559793862541555, −6.927565314040800, −6.459266994455567, −5.871572263882324, −5.524990368784235, −4.816582061549780, −4.106793762806525, −3.650479561329907, −3.216298539228505, −2.418996084325432, −2.023818821416571, −1.202146085312142, −0.6227786002808848,
0.6227786002808848, 1.202146085312142, 2.023818821416571, 2.418996084325432, 3.216298539228505, 3.650479561329907, 4.106793762806525, 4.816582061549780, 5.524990368784235, 5.871572263882324, 6.459266994455567, 6.927565314040800, 7.559793862541555, 7.981249882829842, 8.555788148368428, 9.021712600168147, 9.420484116361194, 9.998754573759118, 10.21284858614106, 11.09641979959068, 11.29557384045620, 11.79307913860414, 12.67883749329966, 12.87001334914578, 13.35956634665767
