| L(s) = 1 | + (−2.44 − 1.41i)2-s + (3.99 + 6.92i)4-s + (7.97 − 4.60i)5-s + (−27.2 + 47.2i)7-s − 22.6i·8-s − 26.0·10-s + (34.3 + 19.8i)11-s + (−107. − 186. i)13-s + (133. − 77.1i)14-s + (−32.0 + 55.4i)16-s + 28.2i·17-s − 254.·19-s + (63.7 + 36.8i)20-s + (−56.0 − 97.0i)22-s + (801. − 462. i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.318 − 0.184i)5-s + (−0.556 + 0.964i)7-s − 0.353i·8-s − 0.260·10-s + (0.283 + 0.163i)11-s + (−0.638 − 1.10i)13-s + (0.681 − 0.393i)14-s + (−0.125 + 0.216i)16-s + 0.0976i·17-s − 0.704·19-s + (0.159 + 0.0920i)20-s + (−0.115 − 0.200i)22-s + (1.51 − 0.875i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2008958067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2008958067\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.44 + 1.41i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-7.97 + 4.60i)T + (312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (27.2 - 47.2i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-34.3 - 19.8i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (107. + 186. i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 - 28.2iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 254.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-801. + 462. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (1.16e3 + 672. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (588. + 1.01e3i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 2.34e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (1.25e3 - 726. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (837. - 1.45e3i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (2.87e3 + 1.66e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 - 1.74e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (709. - 409. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (393. - 682. i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (1.42e3 + 2.47e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 7.90e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 7.11e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-1.73e3 + 3.00e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (996. + 575. i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + 1.07e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (4.98e3 - 8.63e3i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
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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
−11.67122680763188498093976212416, −10.56194184894103124609509602276, −9.547649152978127748430078526344, −8.861558154636838606347031689796, −7.66359067579579335905545687218, −6.35288848028217649174576634986, −5.15669432275966114804094319915, −3.28135416269961450335083021817, −2.01184541473869320880390228628, −0.088888185955104982266063468310,
1.70035321516695241219496604682, 3.58845529223399943831746453426, 5.17391955478848289890213021080, 6.76450054630091061041283340742, 7.12085325586985075994172141967, 8.709203332036485648526146100805, 9.579456297712314060416143514952, 10.48302867601768463882267130022, 11.41046220677126944928942752632, 12.72376247643983986350725226299
