| L(s) = 1 | + (0.930 + 0.474i)2-s + (−2.78 + 0.440i)3-s + (−0.533 − 0.734i)4-s + (−1.77 − 1.35i)5-s + (−2.79 − 0.909i)6-s + (−0.0860 + 0.543i)7-s + (−0.475 − 3.00i)8-s + (4.68 − 1.52i)9-s + (−1.01 − 2.10i)10-s + (1.80 + 1.80i)12-s + (−1.47 + 2.89i)13-s + (−0.337 + 0.464i)14-s + (5.54 + 2.99i)15-s + (0.419 − 1.29i)16-s + (2.57 + 5.04i)17-s + (5.08 + 0.805i)18-s + ⋯ |
| L(s) = 1 | + (0.658 + 0.335i)2-s + (−1.60 + 0.254i)3-s + (−0.266 − 0.367i)4-s + (−0.794 − 0.607i)5-s + (−1.14 − 0.371i)6-s + (−0.0325 + 0.205i)7-s + (−0.168 − 1.06i)8-s + (1.56 − 0.507i)9-s + (−0.319 − 0.666i)10-s + (0.522 + 0.522i)12-s + (−0.408 + 0.801i)13-s + (−0.0902 + 0.124i)14-s + (1.43 + 0.772i)15-s + (0.104 − 0.322i)16-s + (0.623 + 1.22i)17-s + (1.19 + 0.189i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.479615 + 0.414802i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.479615 + 0.414802i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.77 + 1.35i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.930 - 0.474i)T + (1.17 + 1.61i)T^{2} \) |
| 3 | \( 1 + (2.78 - 0.440i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (0.0860 - 0.543i)T + (-6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (1.47 - 2.89i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-2.57 - 5.04i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 0.914i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.803 - 0.803i)T - 23iT^{2} \) |
| 29 | \( 1 + (-3.44 + 2.50i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.509 + 1.56i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.945 + 0.149i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (5.25 - 7.23i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.55 - 2.55i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.636 - 4.02i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-6.27 - 3.19i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (3.97 + 5.47i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.75 - 2.84i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (2.62 + 2.62i)T + 67iT^{2} \) |
| 71 | \( 1 + (-2.11 + 6.51i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-9.96 - 1.57i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-1.28 - 3.96i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (10.0 - 5.14i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + 3.64iT - 89T^{2} \) |
| 97 | \( 1 + (7.56 - 14.8i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−10.98778172074156100017211824960, −10.10925827695138327829192203018, −9.332254172013976360733476643638, −8.070880848565579303811685688514, −6.87383330805491990209391112819, −6.05498722010257464372017381095, −5.32705885317437699370540730153, −4.53929432911938292830153839072, −3.84522558040415414040387757693, −1.11451503275913101587675395250,
0.43539766217801111047051886209, 2.79826840886023392906380222339, 3.90831080680485163964832070786, 5.00683424873486954397466184908, 5.54339770786360219277348760944, 6.91843192850281786390327001946, 7.42780800903405900152622551443, 8.547362965839586027957342066081, 10.07055123650754002468357374135, 10.76286119626935664177867450831
