| L(s) = 1 | + 10·5-s − 309. i·7-s + 960. i·11-s − 1.46e3·13-s + 4.76e3·17-s + 7.52e3i·19-s − 1.04e4i·23-s − 1.55e4·25-s + 2.54e4·29-s − 4.18e4i·31-s − 3.09e3i·35-s − 1.99e3·37-s − 2.93e4·41-s − 2.15e4i·43-s − 7.56e3i·47-s + ⋯ |
| L(s) = 1 | + 0.0800·5-s − 0.903i·7-s + 0.721i·11-s − 0.667·13-s + 0.970·17-s + 1.09i·19-s − 0.860i·23-s − 0.993·25-s + 1.04·29-s − 1.40i·31-s − 0.0722i·35-s − 0.0393·37-s − 0.426·41-s − 0.270i·43-s − 0.0728i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.2622046021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2622046021\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 10T + 1.56e4T^{2} \) |
| 7 | \( 1 + 309. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 960. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.46e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 4.76e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 7.52e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.04e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.54e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 4.18e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.99e3T + 2.56e9T^{2} \) |
| 41 | \( 1 + 2.93e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 2.15e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 7.56e3iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.92e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 7.84e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.09e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.94e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 5.32e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.88e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 3.10e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 2.04e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 3.10e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.45e6T + 8.32e11T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.718761201161987101348127668191, −8.216116399895786319108095395298, −7.60937701032230716300952885254, −6.70536594872214477071005980620, −5.65818644223483713143207135738, −4.54503214509918485835013263275, −3.73617730438942251416842374580, −2.41600329744770126033256678430, −1.25591662480673276858272349145, −0.05340908217339303632296040367,
1.29373837878923311736647593658, 2.57146549740577268030815949418, 3.40463832847471777995266124285, 4.86748051499853641043309778178, 5.58647841623361020539511342101, 6.54511847673326421012661054754, 7.61798137123829767011011055924, 8.504569176886728919234945613003, 9.311997152699814113906597715465, 10.09476348211837089996993303214
