| L(s) = 1 | + (1.52 − 3.69i)2-s + (4.76 + 7.63i)3-s + (−11.3 − 11.2i)4-s + (11.0 − 19.1i)5-s + (35.5 − 5.96i)6-s + (82.7 − 47.7i)7-s + (−59.0 + 24.6i)8-s + (−35.5 + 72.7i)9-s + (−54.0 − 70.2i)10-s + (−18.9 + 10.9i)11-s + (32.1 − 140. i)12-s + (−63.1 + 109. i)13-s + (−50.2 − 379. i)14-s + (199. − 6.89i)15-s + (0.961 + 255. i)16-s − 283.·17-s + ⋯ |
| L(s) = 1 | + (0.381 − 0.924i)2-s + (0.529 + 0.848i)3-s + (−0.708 − 0.705i)4-s + (0.442 − 0.767i)5-s + (0.986 − 0.165i)6-s + (1.68 − 0.975i)7-s + (−0.922 + 0.385i)8-s + (−0.438 + 0.898i)9-s + (−0.540 − 0.702i)10-s + (−0.156 + 0.0903i)11-s + (0.223 − 0.974i)12-s + (−0.373 + 0.646i)13-s + (−0.256 − 1.93i)14-s + (0.885 − 0.0306i)15-s + (0.00375 + 0.999i)16-s − 0.982·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.873i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.485 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.79465 - 1.05557i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79465 - 1.05557i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.52 + 3.69i)T \) |
| 3 | \( 1 + (-4.76 - 7.63i)T \) |
| good | 5 | \( 1 + (-11.0 + 19.1i)T + (-312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (-82.7 + 47.7i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (18.9 - 10.9i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (63.1 - 109. i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + 283.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 323. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (-198. - 114. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (604. + 1.04e3i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-718. - 414. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 318.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (164. - 284. i)T + (-1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (179. - 103. i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (1.06e3 - 613. i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 2.83e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-1.27e3 - 737. i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-936. - 1.62e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-214. - 123. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 4.30e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 3.01e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-6.22e3 + 3.59e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-2.87e3 + 1.66e3i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + 1.54e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (2.91e3 + 5.05e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−15.07793469976111346000522661502, −14.13657084464396385093454203571, −13.35354148106591791088424975648, −11.58966865187078553820424523351, −10.61096962478871131523462593704, −9.395456513372724166119931030880, −8.166475788914881573201727375704, −5.06361741407098228571245424681, −4.23191442437997337788295785348, −1.78236657500407336433791857894,
2.53768280405582404964005322239, 5.17226941174125247687360751797, 6.68664915500344077050197863785, 7.974303155291473784461376766364, 8.952186635593864995146148182425, 11.27432323606327890177863859947, 12.62648112960925729408072143690, 13.85162830213838538158331635689, 14.73836921888655025963551358687, 15.32444568245852618702497507870
