| L(s) = 1 | + 2-s + 3·3-s + 4-s + 3·6-s + 8-s + 6·9-s + 3·12-s + 7·13-s + 16-s − 7·17-s + 6·18-s + 19-s + 8·23-s + 3·24-s − 5·25-s + 7·26-s + 9·27-s − 9·29-s + 2·31-s + 32-s − 7·34-s + 6·36-s + 3·37-s + 38-s + 21·39-s + 10·41-s − 10·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.22·6-s + 0.353·8-s + 2·9-s + 0.866·12-s + 1.94·13-s + 1/4·16-s − 1.69·17-s + 1.41·18-s + 0.229·19-s + 1.66·23-s + 0.612·24-s − 25-s + 1.37·26-s + 1.73·27-s − 1.67·29-s + 0.359·31-s + 0.176·32-s − 1.20·34-s + 36-s + 0.493·37-s + 0.162·38-s + 3.36·39-s + 1.56·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.451465502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.451465502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.336384672381450914953441469658, −7.68518050899319822002708251361, −6.89309234270293758384664065378, −6.26555728404467171168821694932, −5.25793582163031407488407859416, −4.15039718532399784162258574085, −3.81474771643367942690779784838, −2.99633193353632251635025171957, −2.20065862699872789756554872639, −1.34198304852947418622224891762,
1.34198304852947418622224891762, 2.20065862699872789756554872639, 2.99633193353632251635025171957, 3.81474771643367942690779784838, 4.15039718532399784162258574085, 5.25793582163031407488407859416, 6.26555728404467171168821694932, 6.89309234270293758384664065378, 7.68518050899319822002708251361, 8.336384672381450914953441469658
