| L(s) = 1 | + (0.291 + 0.323i)2-s + (0.0345 + 0.549i)3-s + (0.189 − 1.80i)4-s + (−0.0562 + 0.133i)5-s + (−0.167 + 0.171i)6-s + (1.88 + 0.158i)7-s + (1.34 − 0.976i)8-s + (2.67 − 0.338i)9-s + (−0.0597 + 0.0208i)10-s + (−0.171 − 0.114i)11-s + (0.995 + 0.0417i)12-s + (−1.98 + 5.33i)13-s + (0.498 + 0.657i)14-s + (−0.0755 − 0.0262i)15-s + (−2.83 − 0.602i)16-s + (−4.09 − 5.89i)17-s + ⋯ |
| L(s) = 1 | + (0.206 + 0.229i)2-s + (0.0199 + 0.317i)3-s + (0.0945 − 0.900i)4-s + (−0.0251 + 0.0598i)5-s + (−0.0685 + 0.0700i)6-s + (0.712 + 0.0598i)7-s + (0.475 − 0.345i)8-s + (0.891 − 0.112i)9-s + (−0.0189 + 0.00658i)10-s + (−0.0518 − 0.0344i)11-s + (0.287 + 0.0120i)12-s + (−0.550 + 1.48i)13-s + (0.133 + 0.175i)14-s + (−0.0194 − 0.00678i)15-s + (−0.708 − 0.150i)16-s + (−0.992 − 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34437 + 0.0188103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34437 + 0.0188103i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (1.77 + 12.1i)T \) |
| good | 2 | \( 1 + (-0.291 - 0.323i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.0345 - 0.549i)T + (-2.97 + 0.375i)T^{2} \) |
| 5 | \( 1 + (0.0562 - 0.133i)T + (-3.49 - 3.57i)T^{2} \) |
| 7 | \( 1 + (-1.88 - 0.158i)T + (6.90 + 1.16i)T^{2} \) |
| 11 | \( 1 + (0.171 + 0.114i)T + (4.26 + 10.1i)T^{2} \) |
| 13 | \( 1 + (1.98 - 5.33i)T + (-9.84 - 8.49i)T^{2} \) |
| 17 | \( 1 + (4.09 + 5.89i)T + (-5.92 + 15.9i)T^{2} \) |
| 19 | \( 1 + (-1.09 - 0.796i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (4.15 - 1.84i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (9.19 + 3.64i)T + (21.1 + 19.8i)T^{2} \) |
| 31 | \( 1 + (-6.07 + 1.83i)T + (25.8 - 17.1i)T^{2} \) |
| 37 | \( 1 + (6.73 - 1.13i)T + (34.9 - 12.1i)T^{2} \) |
| 41 | \( 1 + (-6.27 - 1.61i)T + (35.9 + 19.7i)T^{2} \) |
| 43 | \( 1 + (8.33 - 0.699i)T + (42.3 - 7.17i)T^{2} \) |
| 47 | \( 1 + (-2.65 - 9.51i)T + (-40.2 + 24.3i)T^{2} \) |
| 53 | \( 1 + (1.52 + 2.40i)T + (-22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (-1.29 - 3.97i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.04 + 2.50i)T + (36.8 - 48.5i)T^{2} \) |
| 67 | \( 1 + (-9.12 - 1.15i)T + (64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (2.54 - 3.66i)T + (-24.7 - 66.5i)T^{2} \) |
| 73 | \( 1 + (0.562 + 1.19i)T + (-46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (-12.1 + 11.4i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-9.55 + 5.25i)T + (44.4 - 70.0i)T^{2} \) |
| 89 | \( 1 + (9.34 + 5.66i)T + (41.2 + 78.8i)T^{2} \) |
| 97 | \( 1 + (3.38 - 4.45i)T + (-26.0 - 93.4i)T^{2} \) |
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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
−13.31012769700369266426328102178, −11.74598279228696584778817693065, −11.09753317962484834531007985833, −9.790435887068954341023215792668, −9.246356652338019829454150548793, −7.47205721602801949400589354292, −6.60549943964121356452754585497, −5.08943953922996863573732984670, −4.30849519860851710994124766696, −1.87429366597576115431120147761,
2.12071911935591906231721041013, 3.82666928691803559169287369952, 5.03086500603396595483829036740, 6.78596868379754137616498215123, 7.85786352718294376220442388482, 8.499735589618259980629167234893, 10.22802224489660674390370229752, 11.03195032689895705544322349653, 12.35355359517410905675791466660, 12.76119712403460020938799559150
