| L(s) = 1 | + (−3.64 − 1.65i)2-s + 12.7·3-s + (10.5 + 12.0i)4-s − 20.4·5-s + (−46.3 − 21.1i)6-s − 9.19·7-s + (−18.2 − 61.3i)8-s + 81.2·9-s + (74.4 + 33.8i)10-s − 123.·11-s + (133. + 153. i)12-s + 161. i·13-s + (33.4 + 15.2i)14-s − 260.·15-s + (−35.1 + 253. i)16-s + 238. i·17-s + ⋯ |
| L(s) = 1 | + (−0.910 − 0.414i)2-s + 1.41·3-s + (0.656 + 0.754i)4-s − 0.817·5-s + (−1.28 − 0.586i)6-s − 0.187·7-s + (−0.285 − 0.958i)8-s + 1.00·9-s + (0.744 + 0.338i)10-s − 1.02·11-s + (0.929 + 1.06i)12-s + 0.958i·13-s + (0.170 + 0.0777i)14-s − 1.15·15-s + (−0.137 + 0.990i)16-s + 0.823i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5288031968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5288031968\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.64 + 1.65i)T \) |
| 41 | \( 1 + (1.67e3 + 131. i)T \) |
| good | 3 | \( 1 - 12.7T + 81T^{2} \) |
| 5 | \( 1 + 20.4T + 625T^{2} \) |
| 7 | \( 1 + 9.19T + 2.40e3T^{2} \) |
| 11 | \( 1 + 123.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 161. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 238. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 236.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 267. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 419. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 735. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 394.T + 1.87e6T^{2} \) |
| 43 | \( 1 - 3.17e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 321.T + 4.87e6T^{2} \) |
| 53 | \( 1 - 3.56e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 2.64e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.73e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 3.66e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.83e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 561.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 2.40e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 8.04e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 8.33e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 7.95e3iT - 8.85e7T^{2} \) |
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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
−12.49347898389666642852817103800, −11.35136089100850118328230888935, −10.30221560590204087484437168689, −9.299834186767686527304163753990, −8.306697001634049855609625357625, −7.898427548061565534811410308170, −6.65688862256583815167556417729, −4.17806279119140563645081475672, −3.10484931065327672403815109042, −1.93667224796856587270589712277,
0.20826879596979836897283578085, 2.28775518792314745851942470807, 3.44490382798233161559456377738, 5.32302090595875692173663146193, 7.06859213174223484527317074172, 7.939546370125672176465803765884, 8.433144429200600241887641807342, 9.555602988514567666399632691408, 10.44851924245458529532136433376, 11.61231787358721509679420038568
