| L(s) = 1 | + 2.64·2-s − 25·4-s − 150.·8-s − 799.·11-s + 401·16-s − 2.11e3·22-s − 1.10e3·23-s − 3.12e3·25-s + 5.38e3·29-s + 5.88e3·32-s + 8.88e3·37-s − 1.17e4·43-s + 1.99e4·44-s − 2.92e3·46-s − 8.26e3·50-s − 3.27e4·53-s + 1.42e4·58-s + 2.74e3·64-s + 6.93e4·67-s + 8.49e4·71-s + 2.35e4·74-s + 8.01e4·79-s − 3.10e4·86-s + 1.20e5·88-s + 2.76e4·92-s + 7.81e4·100-s − 8.65e4·106-s + ⋯ |
| L(s) = 1 | + 0.467·2-s − 0.781·4-s − 0.833·8-s − 1.99·11-s + 0.391·16-s − 0.931·22-s − 0.435·23-s − 25-s + 1.18·29-s + 1.01·32-s + 1.06·37-s − 0.968·43-s + 1.55·44-s − 0.203·46-s − 0.467·50-s − 1.59·53-s + 0.556·58-s + 0.0837·64-s + 1.88·67-s + 1.99·71-s + 0.499·74-s + 1.44·79-s − 0.453·86-s + 1.65·88-s + 0.340·92-s + 0.781·100-s − 0.748·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.184525928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.184525928\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.64T + 32T^{2} \) |
| 5 | \( 1 + 3.12e3T^{2} \) |
| 11 | \( 1 + 799.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.58e9T^{2} \) |
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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
−10.19291046148522459549216267902, −9.576159980438837808377951025933, −8.281773270098613762436877864159, −7.84704461165404591598995568846, −6.33687991319700138749495097124, −5.34497644825417837060113296342, −4.66678012638136598455842485655, −3.44330473172198677993418964807, −2.36312681513080519248858569483, −0.50584985409870227403924018514,
0.50584985409870227403924018514, 2.36312681513080519248858569483, 3.44330473172198677993418964807, 4.66678012638136598455842485655, 5.34497644825417837060113296342, 6.33687991319700138749495097124, 7.84704461165404591598995568846, 8.281773270098613762436877864159, 9.576159980438837808377951025933, 10.19291046148522459549216267902
