L(s) = 1 | + i·2-s + (0.950 − 1.44i)3-s − 4-s − 0.575·5-s + (1.44 + 0.950i)6-s + (2.10 − 1.60i)7-s − i·8-s + (−1.19 − 2.75i)9-s − 0.575i·10-s − 6.32i·11-s + (−0.950 + 1.44i)12-s + 2.57i·13-s + (1.60 + 2.10i)14-s + (−0.547 + 0.834i)15-s + 16-s − 6.33·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.548 − 0.836i)3-s − 0.5·4-s − 0.257·5-s + (0.591 + 0.387i)6-s + (0.793 − 0.608i)7-s − 0.353i·8-s + (−0.398 − 0.917i)9-s − 0.182i·10-s − 1.90i·11-s + (−0.274 + 0.418i)12-s + 0.714i·13-s + (0.429 + 0.561i)14-s + (−0.141 + 0.215i)15-s + 0.250·16-s − 1.53·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0728 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0728 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06910 - 0.993836i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06910 - 0.993836i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.950 + 1.44i)T \) |
| 7 | \( 1 + (-2.10 + 1.60i)T \) |
| 23 | \( 1 + iT \) |
good | 5 | \( 1 + 0.575T + 5T^{2} \) |
| 11 | \( 1 + 6.32iT - 11T^{2} \) |
| 13 | \( 1 - 2.57iT - 13T^{2} \) |
| 17 | \( 1 + 6.33T + 17T^{2} \) |
| 19 | \( 1 - 4.31iT - 19T^{2} \) |
| 29 | \( 1 + 10.1iT - 29T^{2} \) |
| 31 | \( 1 - 1.82iT - 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 6.83T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 - 4.99iT - 53T^{2} \) |
| 59 | \( 1 - 2.62T + 59T^{2} \) |
| 61 | \( 1 + 6.43iT - 61T^{2} \) |
| 67 | \( 1 + 1.55T + 67T^{2} \) |
| 71 | \( 1 + 4.59iT - 71T^{2} \) |
| 73 | \( 1 + 13.0iT - 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 2.97T + 83T^{2} \) |
| 89 | \( 1 + 3.91T + 89T^{2} \) |
| 97 | \( 1 - 17.9iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.434258928776950768708287458553, −8.664351888506938858452614149186, −8.051635516989064722968295536783, −7.49042531132408021146666263747, −6.37173828564397123954151022897, −5.90302963682643494362112309408, −4.38329882609328510708769252754, −3.63443762674381440315327329040, −2.12393657281033018599994447317, −0.62298771667593081792055393717,
1.96535910683241567426867747970, 2.67101660622927581922338708479, 4.07758030341295461714093019708, 4.68936667476581195042636019577, 5.42386235674545479560730565377, 7.09441196026731329882755153702, 7.915482594460002138897664625706, 8.898394265866977676825391434524, 9.316129828690953149782154721651, 10.25524937224167018517063503090