| L(s) = 1 | + (−1 + 2.82i)3-s + (−7.00 − 5.65i)9-s + (−7 + 8.48i)11-s + 2·17-s − 16.9i·19-s − 25·25-s + (23.0 − 14.1i)27-s + (−17 − 28.2i)33-s + 46·41-s − 84.8i·43-s − 49·49-s + (−2 + 5.65i)51-s + (48 + 16.9i)57-s − 84.8i·59-s − 62·67-s + ⋯ |
| L(s) = 1 | + (−0.333 + 0.942i)3-s + (−0.777 − 0.628i)9-s + (−0.636 + 0.771i)11-s + 0.117·17-s − 0.893i·19-s − 25-s + (0.851 − 0.523i)27-s + (−0.515 − 0.857i)33-s + 1.12·41-s − 1.97i·43-s − 0.999·49-s + (−0.0392 + 0.110i)51-s + (0.842 + 0.297i)57-s − 1.43i·59-s − 0.925·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7811218784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7811218784\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 2.82i)T \) |
| 11 | \( 1 + (7 - 8.48i)T \) |
| good | 5 | \( 1 + 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 2T + 289T^{2} \) |
| 19 | \( 1 + 16.9iT - 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 46T + 1.68e3T^{2} \) |
| 43 | \( 1 + 84.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 84.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 + 62T + 4.48e3T^{2} \) |
| 71 | \( 1 + 5.04e3T^{2} \) |
| 73 | \( 1 - 33.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 - 158T + 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 94T + 9.40e3T^{2} \) |
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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
−9.658932997433688436744039933481, −8.969768151467693515926175508191, −7.967563471673870768092351759060, −7.04573539665807681567539622219, −6.00049506758274207334487274395, −5.13966547196095560216471796280, −4.40884145850301233134745216163, −3.38208542545633506995377187646, −2.19852038184257241638877094113, −0.27829681406431511149000545911,
1.09899371293680560056646668553, 2.33805510264295866628275199535, 3.42208970906318060223448768761, 4.79618872750681829973920187337, 5.86570806460058336494951993635, 6.26071501676796686012561304239, 7.57970708327978390829330269559, 7.917796755522532642344571557887, 8.861543262216822236440766527062, 9.924469846942041938204823725748
