| L(s) = 1 | + 3-s − 2·4-s − 5-s − 4.23·7-s − 2·9-s + 1.85·11-s − 2·12-s + 3.23·13-s − 15-s + 4·16-s − 1.85·17-s − 6.47·19-s + 2·20-s − 4.23·21-s − 7.85·23-s + 25-s − 5·27-s + 8.47·28-s + 5.47·31-s + 1.85·33-s + 4.23·35-s + 4·36-s − 5.76·37-s + 3.23·39-s − 9.70·41-s + 3.61·43-s − 3.70·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 1.60·7-s − 0.666·9-s + 0.559·11-s − 0.577·12-s + 0.897·13-s − 0.258·15-s + 16-s − 0.449·17-s − 1.48·19-s + 0.447·20-s − 0.924·21-s − 1.63·23-s + 0.200·25-s − 0.962·27-s + 1.60·28-s + 0.982·31-s + 0.322·33-s + 0.716·35-s + 0.666·36-s − 0.947·37-s + 0.518·39-s − 1.51·41-s + 0.551·43-s − 0.559·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6139648646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6139648646\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 7.85T + 23T^{2} \) |
| 31 | \( 1 - 5.47T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 + 9.70T + 41T^{2} \) |
| 43 | \( 1 - 3.61T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 8.56T + 71T^{2} \) |
| 73 | \( 1 + 2.32T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 - 6.70T + 89T^{2} \) |
| 97 | \( 1 + 1.94T + 97T^{2} \) |
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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
−8.331127389313440577377472207695, −8.175868242690917280480046892129, −6.69280298464385113273161308301, −6.36818524519496707801981969281, −5.51197108514426980058893554778, −4.33042201889232601851480557248, −3.71084203234995699264682797258, −3.27439670579290600344577135344, −2.08289937260327230825684404471, −0.41242305274089011301832990749,
0.41242305274089011301832990749, 2.08289937260327230825684404471, 3.27439670579290600344577135344, 3.71084203234995699264682797258, 4.33042201889232601851480557248, 5.51197108514426980058893554778, 6.36818524519496707801981969281, 6.69280298464385113273161308301, 8.175868242690917280480046892129, 8.331127389313440577377472207695
