| L(s) = 1 | − 2-s − 3-s + 4-s + 3.78·5-s + 6-s − 3.09·7-s − 8-s + 9-s − 3.78·10-s − 1.99·11-s − 12-s − 13-s + 3.09·14-s − 3.78·15-s + 16-s − 7.51·17-s − 18-s + 6.13·19-s + 3.78·20-s + 3.09·21-s + 1.99·22-s + 6.69·23-s + 24-s + 9.32·25-s + 26-s − 27-s − 3.09·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.69·5-s + 0.408·6-s − 1.17·7-s − 0.353·8-s + 0.333·9-s − 1.19·10-s − 0.600·11-s − 0.288·12-s − 0.277·13-s + 0.827·14-s − 0.977·15-s + 0.250·16-s − 1.82·17-s − 0.235·18-s + 1.40·19-s + 0.846·20-s + 0.675·21-s + 0.424·22-s + 1.39·23-s + 0.204·24-s + 1.86·25-s + 0.196·26-s − 0.192·27-s − 0.585·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.231694329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.231694329\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 3.78T + 5T^{2} \) |
| 7 | \( 1 + 3.09T + 7T^{2} \) |
| 11 | \( 1 + 1.99T + 11T^{2} \) |
| 17 | \( 1 + 7.51T + 17T^{2} \) |
| 19 | \( 1 - 6.13T + 19T^{2} \) |
| 23 | \( 1 - 6.69T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 + 0.695T + 37T^{2} \) |
| 41 | \( 1 - 6.91T + 41T^{2} \) |
| 43 | \( 1 - 5.14T + 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 + 5.97T + 53T^{2} \) |
| 59 | \( 1 - 5.70T + 59T^{2} \) |
| 61 | \( 1 + 9.33T + 61T^{2} \) |
| 67 | \( 1 - 0.628T + 67T^{2} \) |
| 71 | \( 1 + 6.41T + 71T^{2} \) |
| 73 | \( 1 + 5.18T + 73T^{2} \) |
| 79 | \( 1 + 5.82T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 2.07T + 97T^{2} \) |
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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
−7.71504220104574234061422552465, −6.89959487276536765787058484944, −6.51890324162544722623671903003, −5.94228329333435019784925355165, −5.23064013685397691673065866602, −4.55573846675213279752752149208, −3.01985141272490055549232708406, −2.66123995480419662857752909252, −1.64671562415157740440563440307, −0.63001145263019747997391466181,
0.63001145263019747997391466181, 1.64671562415157740440563440307, 2.66123995480419662857752909252, 3.01985141272490055549232708406, 4.55573846675213279752752149208, 5.23064013685397691673065866602, 5.94228329333435019784925355165, 6.51890324162544722623671903003, 6.89959487276536765787058484944, 7.71504220104574234061422552465
