| L(s) = 1 | + (1.42 − 0.977i)3-s + 2.48·5-s − 4.85i·7-s + (1.08 − 2.79i)9-s − 3.10i·11-s − i·13-s + (3.55 − 2.42i)15-s + 5.59i·17-s + (−4.75 − 6.94i)21-s + 8.88·23-s + 1.17·25-s + (−1.17 − 5.06i)27-s + 1.06·29-s + 4i·31-s + (−3.03 − 4.44i)33-s + ⋯ |
| L(s) = 1 | + (0.825 − 0.564i)3-s + 1.11·5-s − 1.83i·7-s + (0.362 − 0.931i)9-s − 0.936i·11-s − 0.277i·13-s + (0.917 − 0.627i)15-s + 1.35i·17-s + (−1.03 − 1.51i)21-s + 1.85·23-s + 0.235·25-s + (−0.226 − 0.974i)27-s + 0.196·29-s + 0.718i·31-s + (−0.528 − 0.773i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.083947778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083947778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.42 + 0.977i)T \) |
| 13 | \( 1 + iT \) |
| good | 5 | \( 1 - 2.48T + 5T^{2} \) |
| 7 | \( 1 + 4.85iT - 7T^{2} \) |
| 11 | \( 1 + 3.10iT - 11T^{2} \) |
| 17 | \( 1 - 5.59iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8.88T + 23T^{2} \) |
| 29 | \( 1 - 1.06T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 - 11.5iT - 37T^{2} \) |
| 41 | \( 1 + 5.95iT - 41T^{2} \) |
| 43 | \( 1 + 5.21T + 43T^{2} \) |
| 47 | \( 1 + 1.15T + 47T^{2} \) |
| 53 | \( 1 - 5.15T + 53T^{2} \) |
| 59 | \( 1 - 7.01iT - 59T^{2} \) |
| 61 | \( 1 + 8.07iT - 61T^{2} \) |
| 67 | \( 1 + 5.71T + 67T^{2} \) |
| 71 | \( 1 + 1.15T + 71T^{2} \) |
| 73 | \( 1 + 8.07T + 73T^{2} \) |
| 79 | \( 1 + 1.71iT - 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 - 8.25iT - 89T^{2} \) |
| 97 | \( 1 + 3.71T + 97T^{2} \) |
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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
−8.583926567270844705295778617471, −8.046593468505045162304723371026, −7.04273702964088844296921814156, −6.64265989359746615286143062112, −5.76671109007499914670074768133, −4.63686399066654584960214668505, −3.60141021242229716768198629110, −3.00379150355174964598770910861, −1.61725060951436410194255252338, −0.956883532741760243614191156416,
1.81693505406013742478801402221, 2.46117451139719164823773134489, 3.08319572392581474196946653821, 4.57239278658085061888524224276, 5.18264150767426370786303601161, 5.82286892280487529247525488946, 6.87818234369011270603820936700, 7.66815371776125942350021866080, 8.829099364672420547316295394387, 9.121703535981693085000461483788
