| L(s) = 1 | + (0.997 − 0.0713i)2-s + (0.452 − 2.08i)3-s + (0.989 − 0.142i)4-s + (−1.62 − 1.53i)5-s + (0.303 − 2.10i)6-s + (−0.567 + 1.03i)7-s + (0.977 − 0.212i)8-s + (−1.39 − 0.637i)9-s + (−1.73 − 1.41i)10-s + (1.51 − 1.31i)11-s + (0.151 − 2.12i)12-s + (0.355 − 0.194i)13-s + (−0.491 + 1.07i)14-s + (−3.93 + 2.68i)15-s + (0.959 − 0.281i)16-s + (0.119 + 0.159i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0504i)2-s + (0.261 − 1.20i)3-s + (0.494 − 0.0711i)4-s + (−0.726 − 0.686i)5-s + (0.123 − 0.860i)6-s + (−0.214 + 0.392i)7-s + (0.345 − 0.0751i)8-s + (−0.465 − 0.212i)9-s + (−0.547 − 0.447i)10-s + (0.456 − 0.395i)11-s + (0.0438 − 0.613i)12-s + (0.0985 − 0.0538i)13-s + (−0.131 + 0.287i)14-s + (−1.01 + 0.693i)15-s + (0.239 − 0.0704i)16-s + (0.0288 + 0.0386i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34015 - 1.16803i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34015 - 1.16803i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 + 0.0713i)T \) |
| 5 | \( 1 + (1.62 + 1.53i)T \) |
| 23 | \( 1 + (4.01 - 2.62i)T \) |
| good | 3 | \( 1 + (-0.452 + 2.08i)T + (-2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (0.567 - 1.03i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-1.51 + 1.31i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.355 + 0.194i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-0.119 - 0.159i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.301 - 2.09i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-6.77 - 0.973i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-4.28 - 2.75i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-2.08 - 5.58i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (0.0109 + 0.0239i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (6.74 + 1.46i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (3.75 + 3.75i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.474 - 0.258i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-3.65 + 12.4i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (7.12 - 11.0i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (0.104 + 1.46i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (8.17 - 9.43i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (10.6 + 7.93i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (13.4 + 3.95i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (2.62 - 0.979i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-1.06 + 0.684i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.72 - 2.50i)T + (73.3 + 63.5i)T^{2} \) |
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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
−11.97944267262691987550935483951, −11.76857634089421109737606225021, −10.15150926929536027308969436585, −8.654096420462960888716052219208, −7.962113793032765729405584899368, −6.87909017798689012779139607805, −5.89860498643600206949160828430, −4.50311570501599796218936096934, −3.12318955202018156328402993330, −1.42547818070527648138635061031,
2.89758090477722887712956199949, 4.00642082676700069898512257899, 4.60948105954354694934588505990, 6.29044790895886687786315256689, 7.26957570807282249157603037958, 8.507044240658045807106165825987, 9.844963237680973924496088373166, 10.47466714966033129787969502971, 11.45521881438742667037382681982, 12.28779352834600734924871802209
