L(s) = 1 | + (−0.766 + 0.642i)2-s + (1.43 + 0.524i)3-s + (0.173 − 0.984i)4-s + (−1.43 + 0.524i)6-s + (0.500 + 0.866i)8-s + (1.03 + 0.866i)9-s + (0.939 + 1.62i)11-s + (0.766 − 1.32i)12-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s − 1.34·18-s + (−1.76 − 0.642i)22-s + (0.266 + 1.50i)24-s + (−0.939 + 0.342i)25-s + (0.266 + 0.460i)27-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)2-s + (1.43 + 0.524i)3-s + (0.173 − 0.984i)4-s + (−1.43 + 0.524i)6-s + (0.500 + 0.866i)8-s + (1.03 + 0.866i)9-s + (0.939 + 1.62i)11-s + (0.766 − 1.32i)12-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s − 1.34·18-s + (−1.76 − 0.642i)22-s + (0.266 + 1.50i)24-s + (−0.939 + 0.342i)25-s + (0.266 + 0.460i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.409640094\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.409640094\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \) |
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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
−9.115623500276832356290742049954, −8.472898054413535037157185879872, −7.84969618196756758609754496779, −7.05997969144937250215760468822, −6.49647177798377281171472282629, −5.27971444060614805639106407014, −4.34002497035877594857825361573, −3.73699523559635279802040269465, −2.28241363928537793495242523893, −1.75921536093604927199288772808,
1.00744616246581016857511710843, 2.08253507639876977390540712514, 2.91581103553871493582893223457, 3.55829227790061167355420313929, 4.35888337750816924818587293171, 5.97156064650255241956092405162, 6.79895773400317095909959929772, 7.53983572071522912341902369654, 8.306484951970877298009691100134, 8.674632083563868930968133766530