| L(s) = 1 | + 2·2-s + 2·4-s + (−2 − i)5-s + 4i·7-s + (−4 − 2i)10-s − i·11-s + 2i·13-s + 8i·14-s − 4·16-s + 6·17-s + 4i·19-s + (−4 − 2i)20-s − 2i·22-s + 9i·23-s + (3 + 4i)25-s + 4i·26-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + (−0.894 − 0.447i)5-s + 1.51i·7-s + (−1.26 − 0.632i)10-s − 0.301i·11-s + 0.554i·13-s + 2.13i·14-s − 16-s + 1.45·17-s + 0.917i·19-s + (−0.894 − 0.447i)20-s − 0.426i·22-s + 1.87i·23-s + (0.600 + 0.800i)25-s + 0.784i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0830 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0830 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.453481913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.453481913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 29 | \( 1 + (-2 - 5i)T \) |
| good | 2 | \( 1 - 2T + 2T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 9iT - 23T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 9iT - 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 15T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 7iT - 83T^{2} \) |
| 89 | \( 1 - 2iT - 89T^{2} \) |
| 97 | \( 1 - 11T + 97T^{2} \) |
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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
−9.705241165562487214321846287697, −8.950893201578666372078248978392, −8.207370617276444456773597116117, −7.27300529849285495151592315098, −6.13340633299569553057515089107, −5.40613569696574780957360186736, −4.95488288955330625006751518500, −3.59254007917073859749384693798, −3.28478753548417363019912845808, −1.77088019764797899931344369182,
0.64223607711012492924852643119, 2.76150253355720034799778383644, 3.47663691456538662284798495539, 4.37797694144427953366951522619, 4.80012059859727431091523076309, 6.14885502234064744496038314249, 6.83579944325227858511132406863, 7.60973984109287156553467338298, 8.302843257897433570293909223490, 9.738429152327209098155837035283
