| L(s) = 1 | + (−1.92 + 0.408i)2-s + (−0.835 + 0.330i)3-s + (1.69 − 0.754i)4-s + (1.68 + 0.509i)5-s + (1.46 − 0.976i)6-s + (−0.0413 − 1.97i)7-s + (0.228 − 0.166i)8-s + (−1.59 + 1.50i)9-s + (−3.44 − 0.289i)10-s + (0.750 + 5.08i)11-s + (−1.16 + 1.19i)12-s + (1.39 + 2.66i)13-s + (0.884 + 3.77i)14-s + (−1.57 + 0.132i)15-s + (−2.85 + 3.17i)16-s + (4.64 + 2.81i)17-s + ⋯ |
| L(s) = 1 | + (−1.35 + 0.288i)2-s + (−0.482 + 0.190i)3-s + (0.847 − 0.377i)4-s + (0.754 + 0.227i)5-s + (0.600 − 0.398i)6-s + (−0.0156 − 0.745i)7-s + (0.0809 − 0.0588i)8-s + (−0.532 + 0.500i)9-s + (−1.09 − 0.0915i)10-s + (0.226 + 1.53i)11-s + (−0.336 + 0.343i)12-s + (0.386 + 0.738i)13-s + (0.236 + 1.00i)14-s + (−0.407 + 0.0342i)15-s + (−0.713 + 0.792i)16-s + (1.12 + 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.395921 + 0.337716i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.395921 + 0.337716i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-9.72 - 7.51i)T \) |
| good | 2 | \( 1 + (1.92 - 0.408i)T + (1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (0.835 - 0.330i)T + (2.18 - 2.05i)T^{2} \) |
| 5 | \( 1 + (-1.68 - 0.509i)T + (4.16 + 2.76i)T^{2} \) |
| 7 | \( 1 + (0.0413 + 1.97i)T + (-6.99 + 0.293i)T^{2} \) |
| 11 | \( 1 + (-0.750 - 5.08i)T + (-10.5 + 3.17i)T^{2} \) |
| 13 | \( 1 + (-1.39 - 2.66i)T + (-7.41 + 10.6i)T^{2} \) |
| 17 | \( 1 + (-4.64 - 2.81i)T + (7.87 + 15.0i)T^{2} \) |
| 19 | \( 1 + (1.04 + 0.759i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.537 - 5.11i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (-2.23 + 2.69i)T + (-5.43 - 28.4i)T^{2} \) |
| 31 | \( 1 + (2.73 - 2.35i)T + (4.52 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-8.55 - 0.358i)T + (36.8 + 3.09i)T^{2} \) |
| 41 | \( 1 + (-0.386 - 6.13i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (-0.248 + 11.8i)T + (-42.9 - 1.80i)T^{2} \) |
| 47 | \( 1 + (0.684 + 0.339i)T + (28.4 + 37.4i)T^{2} \) |
| 53 | \( 1 + (0.464 - 0.119i)T + (46.4 - 25.5i)T^{2} \) |
| 59 | \( 1 + (0.747 + 2.30i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (7.92 - 6.27i)T + (13.9 - 59.3i)T^{2} \) |
| 67 | \( 1 + (8.73 + 8.20i)T + (4.20 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-0.467 + 0.283i)T + (32.8 - 62.9i)T^{2} \) |
| 73 | \( 1 + (-7.63 - 4.19i)T + (39.1 + 61.6i)T^{2} \) |
| 79 | \( 1 + (-0.643 + 3.37i)T + (-73.4 - 29.0i)T^{2} \) |
| 83 | \( 1 + (7.04 + 0.890i)T + (80.3 + 20.6i)T^{2} \) |
| 89 | \( 1 + (-4.40 + 5.80i)T + (-23.9 - 85.7i)T^{2} \) |
| 97 | \( 1 + (-1.18 + 5.07i)T + (-86.8 - 43.1i)T^{2} \) |
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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
−13.33946519966891711203809347759, −11.92411754037488553810290960735, −10.73009221355716946203327526243, −10.06538685911552863273697482186, −9.362741039321720953740393227115, −8.002564938102580798197904360172, −7.08629827287974284707614126038, −5.95358630610359388085356495562, −4.33274844435536604924064256456, −1.75795524148440114880523893570,
0.909653680269825436516721276241, 2.88583726215884491222670741745, 5.49625654523697290828895856530, 6.19576058593459971966490761274, 7.955151202199417659579526039633, 8.821947158200520682800885494606, 9.539070634040446225407506589940, 10.70946606222336177206296590933, 11.47478116999053359585100816021, 12.46915994835169251186037499146
