L(s) = 1 | + (−1.86 + 0.133i)3-s + (0.540 − 0.841i)4-s + (−0.989 + 0.142i)5-s + (2.48 − 0.357i)9-s + (0.281 + 0.959i)11-s + (−0.898 + 1.64i)12-s + (1.83 − 0.398i)15-s + (−0.415 − 0.909i)16-s + (−0.415 + 0.909i)20-s + (−0.281 − 0.959i)23-s + (0.959 − 0.281i)25-s + (−2.76 + 0.602i)27-s + (−1.29 − 1.49i)31-s + (−0.654 − 1.75i)33-s + (1.04 − 2.28i)36-s + (1.17 − 1.56i)37-s + ⋯ |
L(s) = 1 | + (−1.86 + 0.133i)3-s + (0.540 − 0.841i)4-s + (−0.989 + 0.142i)5-s + (2.48 − 0.357i)9-s + (0.281 + 0.959i)11-s + (−0.898 + 1.64i)12-s + (1.83 − 0.398i)15-s + (−0.415 − 0.909i)16-s + (−0.415 + 0.909i)20-s + (−0.281 − 0.959i)23-s + (0.959 − 0.281i)25-s + (−2.76 + 0.602i)27-s + (−1.29 − 1.49i)31-s + (−0.654 − 1.75i)33-s + (1.04 − 2.28i)36-s + (1.17 − 1.56i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0350 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0350 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4508717245\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4508717245\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.989 - 0.142i)T \) |
| 11 | \( 1 + (-0.281 - 0.959i)T \) |
| 23 | \( 1 + (0.281 + 0.959i)T \) |
good | 2 | \( 1 + (-0.540 + 0.841i)T^{2} \) |
| 3 | \( 1 + (1.86 - 0.133i)T + (0.989 - 0.142i)T^{2} \) |
| 7 | \( 1 + (0.755 + 0.654i)T^{2} \) |
| 13 | \( 1 + (-0.755 + 0.654i)T^{2} \) |
| 17 | \( 1 + (-0.909 - 0.415i)T^{2} \) |
| 19 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 29 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 31 | \( 1 + (1.29 + 1.49i)T + (-0.142 + 0.989i)T^{2} \) |
| 37 | \( 1 + (-1.17 + 1.56i)T + (-0.281 - 0.959i)T^{2} \) |
| 41 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 43 | \( 1 + (0.989 - 0.142i)T^{2} \) |
| 47 | \( 1 + (-0.847 + 0.847i)T - iT^{2} \) |
| 53 | \( 1 + (0.398 + 0.148i)T + (0.755 + 0.654i)T^{2} \) |
| 59 | \( 1 + (-0.983 - 0.449i)T + (0.654 + 0.755i)T^{2} \) |
| 61 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 67 | \( 1 + (-0.334 - 0.613i)T + (-0.540 + 0.841i)T^{2} \) |
| 71 | \( 1 + (1.84 + 0.540i)T + (0.841 + 0.540i)T^{2} \) |
| 73 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 79 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 83 | \( 1 + (-0.281 - 0.959i)T^{2} \) |
| 89 | \( 1 + (-1.19 + 1.37i)T + (-0.142 - 0.989i)T^{2} \) |
| 97 | \( 1 + (-1.28 + 0.959i)T + (0.281 - 0.959i)T^{2} \) |
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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
−10.09008122962085432731855365370, −9.165048399360623283807373580359, −7.54969336323015514913970818220, −7.09581514178569684720010312524, −6.26862903018823360579574888547, −5.59013287977868420111145258984, −4.65606729606758472356011862611, −4.01687210214736779207292086815, −2.04625133138882173199972418923, −0.54647939050369131070633939717,
1.29610639852663413189511207186, 3.24710648075900842549481039691, 4.13116069995146217485599671453, 5.06450603628314098015937372779, 6.03617455799143923306594895902, 6.73865743532793529685918178691, 7.47717270947728388546919156165, 8.169976621830366479995686482248, 9.280288251899863071223172592078, 10.58044477841868199737043082987