| L(s) = 1 | − 2-s − 3-s + 4-s + 4·5-s + 6-s − 3·7-s − 8-s − 2·9-s − 4·10-s − 2·11-s − 12-s + 3·14-s − 4·15-s + 16-s + 3·17-s + 2·18-s + 19-s + 4·20-s + 3·21-s + 2·22-s − 23-s + 24-s + 11·25-s + 5·27-s − 3·28-s − 5·29-s + 4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s − 1.26·10-s − 0.603·11-s − 0.288·12-s + 0.801·14-s − 1.03·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.229·19-s + 0.894·20-s + 0.654·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s + 11/5·25-s + 0.962·27-s − 0.566·28-s − 0.928·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.136089283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.136089283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.081403556814365619620991486748, −7.23769136560338131219122632388, −6.36375886203681583526027515427, −5.98634322619209752201921634912, −5.59820256126613597041086249059, −4.67550199492678466714172566208, −3.10975673351384299574625047165, −2.75001081480826495914336737773, −1.72024833938566855701291965497, −0.62500790976805621272166526863,
0.62500790976805621272166526863, 1.72024833938566855701291965497, 2.75001081480826495914336737773, 3.10975673351384299574625047165, 4.67550199492678466714172566208, 5.59820256126613597041086249059, 5.98634322619209752201921634912, 6.36375886203681583526027515427, 7.23769136560338131219122632388, 8.081403556814365619620991486748
