| L(s) = 1 | + (0.398 − 0.289i)2-s + (0.0979 + 0.0919i)3-s + (−0.542 + 1.67i)4-s + (0.126 + 2.01i)5-s + (0.0657 + 0.00830i)6-s + (−2.56 + 4.04i)7-s + (0.572 + 1.76i)8-s + (−0.187 − 2.97i)9-s + (0.634 + 0.767i)10-s + (2.84 − 2.67i)11-s + (−0.206 + 0.113i)12-s + (2.78 − 5.91i)13-s + (0.148 + 2.35i)14-s + (−0.172 + 0.209i)15-s + (−2.10 − 1.52i)16-s + (2.84 + 4.47i)17-s + ⋯ |
| L(s) = 1 | + (0.282 − 0.204i)2-s + (0.0565 + 0.0530i)3-s + (−0.271 + 0.835i)4-s + (0.0567 + 0.901i)5-s + (0.0268 + 0.00338i)6-s + (−0.971 + 1.53i)7-s + (0.202 + 0.622i)8-s + (−0.0624 − 0.992i)9-s + (0.200 + 0.242i)10-s + (0.857 − 0.805i)11-s + (−0.0597 + 0.0328i)12-s + (0.771 − 1.63i)13-s + (0.0396 + 0.630i)14-s + (−0.0446 + 0.0539i)15-s + (−0.525 − 0.382i)16-s + (0.689 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01978 + 0.603642i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01978 + 0.603642i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-7.02 + 10.0i)T \) |
| good | 2 | \( 1 + (-0.398 + 0.289i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.0979 - 0.0919i)T + (0.188 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-0.126 - 2.01i)T + (-4.96 + 0.626i)T^{2} \) |
| 7 | \( 1 + (2.56 - 4.04i)T + (-2.98 - 6.33i)T^{2} \) |
| 11 | \( 1 + (-2.84 + 2.67i)T + (0.690 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.78 + 5.91i)T + (-8.28 - 10.0i)T^{2} \) |
| 17 | \( 1 + (-2.84 - 4.47i)T + (-7.23 + 15.3i)T^{2} \) |
| 19 | \( 1 + (-0.212 + 0.653i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.988 - 3.04i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.520 + 2.72i)T + (-26.9 - 10.6i)T^{2} \) |
| 31 | \( 1 + (-5.43 - 2.15i)T + (22.5 + 21.2i)T^{2} \) |
| 37 | \( 1 + (0.715 - 1.52i)T + (-23.5 - 28.5i)T^{2} \) |
| 41 | \( 1 + (9.74 + 1.23i)T + (39.7 + 10.1i)T^{2} \) |
| 43 | \( 1 + (-1.49 - 2.35i)T + (-18.3 + 38.9i)T^{2} \) |
| 47 | \( 1 + (-5.31 - 0.670i)T + (45.5 + 11.6i)T^{2} \) |
| 53 | \( 1 + (4.84 + 2.66i)T + (28.3 + 44.7i)T^{2} \) |
| 59 | \( 1 + (5.09 + 3.70i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.927 + 0.870i)T + (3.83 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.686 + 10.9i)T + (-66.4 - 8.39i)T^{2} \) |
| 71 | \( 1 + (6.85 - 10.8i)T + (-30.2 - 64.2i)T^{2} \) |
| 73 | \( 1 + (-3.47 + 5.47i)T + (-31.0 - 66.0i)T^{2} \) |
| 79 | \( 1 + (8.69 - 3.44i)T + (57.5 - 54.0i)T^{2} \) |
| 83 | \( 1 + (-4.14 + 1.06i)T + (72.7 - 39.9i)T^{2} \) |
| 89 | \( 1 + (-5.55 + 1.42i)T + (77.9 - 42.8i)T^{2} \) |
| 97 | \( 1 + (-1.06 + 16.8i)T + (-96.2 - 12.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.94194836833752310326558087445, −12.26545024721579234147664566301, −11.45906906292184232752206157674, −10.12324654547532306894912559582, −8.935573761815202267648674732950, −8.205627177047392619610477385073, −6.46433322821332019556362795086, −5.77231314888879189633478919278, −3.39964608591151037026785802606, −3.13300123525870466119468654134,
1.28800610869513007315850031310, 4.09294274465061099014158867415, 4.82722145035593372451416845897, 6.44755324704851371652422441563, 7.23376466270671437647957396391, 8.940557595906658139403387543531, 9.763119377845790322203492709453, 10.60656759474820383996923463023, 11.96326933333319740546179796724, 13.21427560472921201487673934215
