L(s) = 1 | + 2.37·2-s + 4.44i·3-s − 2.37·4-s − 19.4i·5-s + 10.5i·6-s + 14.9i·7-s − 24.6·8-s + 7.23·9-s − 46.1i·10-s + 31.1i·11-s − 10.5i·12-s − 5.21·13-s + 35.5i·14-s + 86.4·15-s − 39.3·16-s + (68.2 − 16.1i)17-s + ⋯ |
L(s) = 1 | + 0.838·2-s + 0.855i·3-s − 0.296·4-s − 1.73i·5-s + 0.717i·6-s + 0.809i·7-s − 1.08·8-s + 0.267·9-s − 1.45i·10-s + 0.853i·11-s − 0.253i·12-s − 0.111·13-s + 0.678i·14-s + 1.48·15-s − 0.615·16-s + (0.973 − 0.230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.31095 + 0.152878i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31095 + 0.152878i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-68.2 + 16.1i)T \) |
good | 2 | \( 1 - 2.37T + 8T^{2} \) |
| 3 | \( 1 - 4.44iT - 27T^{2} \) |
| 5 | \( 1 + 19.4iT - 125T^{2} \) |
| 7 | \( 1 - 14.9iT - 343T^{2} \) |
| 11 | \( 1 - 31.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 5.21T + 2.19e3T^{2} \) |
| 19 | \( 1 + 28T + 6.85e3T^{2} \) |
| 23 | \( 1 + 167. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 136. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 50.5iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 260. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 183. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 348.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 408.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 108.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 123. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 243.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 42.7iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 875. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 750. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 472.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 376.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 303. iT - 9.12e5T^{2} \) |
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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
−18.42330567947216344792576846589, −16.84098036077514651031132246251, −15.75378030900008580055718880191, −14.61747177278034738770858406032, −12.80989534907016739716398128188, −12.28563232661045129803580875266, −9.709340471103488431391009543474, −8.699953750691681295823385858616, −5.31977091221502005946431867360, −4.37923220276666023327149617093,
3.46918795984664353324196041812, 6.22065577514056024710167293684, 7.56594401686642402426066623063, 10.19791111715273864048033247342, 11.75212381344530589661965658315, 13.42761283186875752180698191079, 14.03930960131214703925058849590, 15.25848146939513783510658829590, 17.43735411737250900906516537789, 18.59641338019204734025794913016