| L(s) = 1 | − 2.73·3-s + 5-s + 2·7-s + 4.46·9-s − 3.46·11-s + 2.73·13-s − 2.73·15-s − 3.46·17-s − 5.46·21-s − 3.46·23-s + 25-s − 3.99·27-s − 3.46·29-s + 1.46·31-s + 9.46·33-s + 2·35-s − 6.73·37-s − 7.46·39-s + 6·41-s − 4.92·43-s + 4.46·45-s + 12.9·47-s − 3·49-s + 9.46·51-s + 10.7·53-s − 3.46·55-s − 6.92·59-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 0.447·5-s + 0.755·7-s + 1.48·9-s − 1.04·11-s + 0.757·13-s − 0.705·15-s − 0.840·17-s − 1.19·21-s − 0.722·23-s + 0.200·25-s − 0.769·27-s − 0.643·29-s + 0.262·31-s + 1.64·33-s + 0.338·35-s − 1.10·37-s − 1.19·39-s + 0.937·41-s − 0.751·43-s + 0.665·45-s + 1.88·47-s − 0.428·49-s + 1.32·51-s + 1.47·53-s − 0.467·55-s − 0.901·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.017786613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.017786613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 6.73T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 + 2.92T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.81532340263642971274507653877, −7.06847443341232265053085334201, −6.36770711573978749714756356731, −5.69885285052986013869079565906, −5.29233199099174526529636884709, −4.59107907412070758447498710536, −3.83203887611251216185402392045, −2.47702267706039017867841470517, −1.64021283516149235189977182489, −0.56286812921932456160399061625,
0.56286812921932456160399061625, 1.64021283516149235189977182489, 2.47702267706039017867841470517, 3.83203887611251216185402392045, 4.59107907412070758447498710536, 5.29233199099174526529636884709, 5.69885285052986013869079565906, 6.36770711573978749714756356731, 7.06847443341232265053085334201, 7.81532340263642971274507653877
