| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (2.50 − 0.544i)3-s + (−0.989 − 0.142i)4-s + (1.09 + 1.95i)5-s + (0.364 + 2.53i)6-s + (0.862 − 0.470i)7-s + (0.212 − 0.977i)8-s + (3.23 − 1.47i)9-s + (−2.02 + 0.950i)10-s + (−4.13 − 3.58i)11-s + (−2.55 + 0.182i)12-s + (−1.66 + 3.04i)13-s + (0.408 + 0.893i)14-s + (3.79 + 4.28i)15-s + (0.959 + 0.281i)16-s + (0.137 + 0.102i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (1.44 − 0.314i)3-s + (−0.494 − 0.0711i)4-s + (0.488 + 0.872i)5-s + (0.148 + 1.03i)6-s + (0.325 − 0.177i)7-s + (0.0751 − 0.345i)8-s + (1.07 − 0.492i)9-s + (−0.639 + 0.300i)10-s + (−1.24 − 1.07i)11-s + (−0.737 + 0.0527i)12-s + (−0.461 + 0.845i)13-s + (0.109 + 0.238i)14-s + (0.980 + 1.10i)15-s + (0.239 + 0.0704i)16-s + (0.0333 + 0.0249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63698 + 0.733027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63698 + 0.733027i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 5 | \( 1 + (-1.09 - 1.95i)T \) |
| 23 | \( 1 + (-2.13 + 4.29i)T \) |
| good | 3 | \( 1 + (-2.50 + 0.544i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-0.862 + 0.470i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (4.13 + 3.58i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.66 - 3.04i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-0.137 - 0.102i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.241 + 1.67i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-5.01 + 0.720i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (8.53 - 5.48i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-4.75 - 1.77i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-1.70 + 3.74i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.12 + 5.15i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (5.74 + 5.74i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.01 + 9.18i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (0.184 + 0.628i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-2.53 - 3.94i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.295 - 0.0211i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-6.31 - 7.29i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.15 - 1.54i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-16.2 + 4.76i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.60 + 4.28i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (8.23 + 5.29i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (3.64 + 9.77i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−12.86018452182599507165539562777, −11.16140803169798495888814108307, −10.20211391482442135678700065987, −9.101414777071472480966200294390, −8.312652609299934259685727352961, −7.43238494716008132405530444563, −6.56735595082040877759693388741, −5.12202327507592344601878976968, −3.40830931286510036825295613134, −2.31904737026924602259355766676,
1.93388243174552360137120013682, 2.97806122152571967941494098311, 4.48191713459646921685505774488, 5.40930728529870897213992764772, 7.74925551257714954430435427063, 8.169414755514691936272398988373, 9.567863233649925883917349530079, 9.645601572286807681189961725068, 10.92467963548521143417452022302, 12.45595922188161141574572216495
