| L(s) = 1 | + (−1.73 + 2i)7-s + 2·13-s + 8i·19-s − 5·25-s + 10.3·31-s − 6.92i·37-s − 10.3·43-s + (−1.00 − 6.92i)49-s − 14·61-s − 3.46·67-s + 13.8i·73-s + 4i·79-s + (−3.46 + 4i)91-s + 13.8i·97-s − 3.46·103-s + ⋯ |
| L(s) = 1 | + (−0.654 + 0.755i)7-s + 0.554·13-s + 1.83i·19-s − 25-s + 1.86·31-s − 1.13i·37-s − 1.58·43-s + (−0.142 − 0.989i)49-s − 1.79·61-s − 0.423·67-s + 1.62i·73-s + 0.450i·79-s + (−0.363 + 0.419i)91-s + 1.40i·97-s − 0.341·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7596064660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7596064660\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.586097141072281232704041263612, −8.197704426598090417948188268205, −7.33241113216993706773176237391, −6.25758803365988851142459280401, −6.02531288589133181974785974644, −5.13990685550451191565453393425, −4.04823026715330309603184572432, −3.39144039885011359499924398264, −2.42252837745485319963731045865, −1.40623653113136167794615730567,
0.22199967097526857505873866772, 1.37282414110226615238196677767, 2.72586901540268924191710107701, 3.39070215828890222431854312023, 4.39454215780990071575395926195, 4.95170363464776228791418730624, 6.20721050083258270892946323074, 6.54045168494148413250046426300, 7.37077694460624816633163990844, 8.102573809432575905587095849560
