| L(s) = 1 | + (−0.766 − 0.944i)2-s + (−2.44 + 2.11i)3-s + (0.106 − 0.509i)4-s + (−0.375 − 1.18i)5-s + (3.86 + 0.684i)6-s + (−1.33 + 3.87i)7-s + (−2.72 + 1.40i)8-s + (1.06 − 7.35i)9-s + (−0.832 + 1.26i)10-s + (1.91 − 0.618i)11-s + (0.815 + 1.46i)12-s + (−4.28 + 1.44i)13-s + (4.67 − 1.71i)14-s + (3.42 + 2.10i)15-s + (2.46 + 1.07i)16-s + (−0.395 − 0.197i)17-s + ⋯ |
| L(s) = 1 | + (−0.542 − 0.667i)2-s + (−1.41 + 1.22i)3-s + (0.0533 − 0.254i)4-s + (−0.168 − 0.530i)5-s + (1.57 + 0.279i)6-s + (−0.504 + 1.46i)7-s + (−0.963 + 0.496i)8-s + (0.355 − 2.45i)9-s + (−0.263 + 0.399i)10-s + (0.576 − 0.186i)11-s + (0.235 + 0.424i)12-s + (−1.18 + 0.401i)13-s + (1.25 − 0.457i)14-s + (0.884 + 0.543i)15-s + (0.615 + 0.269i)16-s + (−0.0958 − 0.0478i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.325343 - 0.236012i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.325343 - 0.236012i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 + (0.409 - 22.4i)T \) |
| good | 2 | \( 1 + (0.766 + 0.944i)T + (-0.410 + 1.95i)T^{2} \) |
| 3 | \( 1 + (2.44 - 2.11i)T + (0.430 - 2.96i)T^{2} \) |
| 5 | \( 1 + (0.375 + 1.18i)T + (-4.08 + 2.87i)T^{2} \) |
| 7 | \( 1 + (1.33 - 3.87i)T + (-5.51 - 4.31i)T^{2} \) |
| 11 | \( 1 + (-1.91 + 0.618i)T + (8.91 - 6.44i)T^{2} \) |
| 13 | \( 1 + (4.28 - 1.44i)T + (10.3 - 7.87i)T^{2} \) |
| 17 | \( 1 + (0.395 + 0.197i)T + (10.2 + 13.5i)T^{2} \) |
| 19 | \( 1 + (-1.39 - 3.13i)T + (-12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.91 + 6.61i)T + (-19.4 - 12.2i)T^{2} \) |
| 29 | \( 1 + (-0.00817 + 0.0683i)T + (-28.1 - 6.83i)T^{2} \) |
| 31 | \( 1 + (0.470 + 5.76i)T + (-30.5 + 5.02i)T^{2} \) |
| 37 | \( 1 + (-0.0289 - 0.0365i)T + (-8.49 + 36.0i)T^{2} \) |
| 41 | \( 1 + (-1.21 + 6.19i)T + (-37.9 - 15.5i)T^{2} \) |
| 43 | \( 1 + (-2.16 + 11.8i)T + (-40.1 - 15.2i)T^{2} \) |
| 47 | \( 1 + (-10.8 + 0.406i)T + (46.8 - 3.52i)T^{2} \) |
| 53 | \( 1 + (-2.92 + 6.55i)T + (-35.4 - 39.4i)T^{2} \) |
| 59 | \( 1 + (0.603 - 1.45i)T + (-41.5 - 41.8i)T^{2} \) |
| 61 | \( 1 + (-2.01 - 3.33i)T + (-28.2 + 54.0i)T^{2} \) |
| 67 | \( 1 + (6.25 - 3.84i)T + (30.3 - 59.7i)T^{2} \) |
| 71 | \( 1 + (-0.902 - 5.72i)T + (-67.5 + 21.8i)T^{2} \) |
| 73 | \( 1 + (8.18 + 7.83i)T + (3.19 + 72.9i)T^{2} \) |
| 79 | \( 1 + (-0.160 - 2.83i)T + (-78.4 + 8.88i)T^{2} \) |
| 83 | \( 1 + (-3.69 - 7.07i)T + (-47.3 + 68.1i)T^{2} \) |
| 89 | \( 1 + (-3.10 - 11.8i)T + (-77.5 + 43.6i)T^{2} \) |
| 97 | \( 1 + (1.02 + 4.36i)T + (-86.7 + 43.3i)T^{2} \) |
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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
−10.61131812390372645626062762048, −10.00754026669882861346990560100, −9.128180721304648622475860843371, −8.851675119510152863930639369491, −6.69439365692555152805304307084, −5.76794409430809009047099566043, −5.23825739178705472681466147824, −4.09794590736068378368408223507, −2.49730352678277231873008948493, −0.44583438190500331537411034602,
0.979261375782716066981756897481, 3.08693994661819468847683649923, 4.69560242345300138569842631771, 6.00951495564287955909757173860, 6.90494610501856175394786713388, 7.26873744781797642970620954649, 7.69953812404066488132043336833, 9.333639839359561282244673232999, 10.35874530344779893219211537705, 11.18002214189633516539077301596
