L(s) = 1 | + (1.70 − 0.323i)3-s − 1.55i·5-s + (2.79 − 1.09i)9-s + 2.53·13-s + (−0.502 − 2.64i)15-s − 4.33i·17-s + 6.83i·19-s + 3.80·23-s + 2.58·25-s + (4.39 − 2.77i)27-s + 8.33i·29-s − 4.89i·31-s + 1.58·37-s + (4.31 − 0.818i)39-s − 7.44i·41-s + ⋯ |
L(s) = 1 | + (0.982 − 0.186i)3-s − 0.695i·5-s + (0.930 − 0.366i)9-s + 0.702·13-s + (−0.129 − 0.683i)15-s − 1.05i·17-s + 1.56i·19-s + 0.794·23-s + 0.516·25-s + (0.845 − 0.533i)27-s + 1.54i·29-s − 0.879i·31-s + 0.260·37-s + (0.690 − 0.131i)39-s − 1.16i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.810273943\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.810273943\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.70 + 0.323i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.55iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 + 4.33iT - 17T^{2} \) |
| 19 | \( 1 - 6.83iT - 19T^{2} \) |
| 23 | \( 1 - 3.80T + 23T^{2} \) |
| 29 | \( 1 - 8.33iT - 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 + 7.44iT - 41T^{2} \) |
| 43 | \( 1 + 6.20iT - 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 8.19T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 4.82T + 83T^{2} \) |
| 89 | \( 1 + 4.33iT - 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−8.738204191559513176282917909837, −8.341101997699372652374092286805, −7.39968277281819967265459983549, −6.78957606281475232990130153656, −5.65839486864211910471222936865, −4.85092451330527943574722344153, −3.84855251123148603809533982105, −3.15405230307011067506379939769, −1.95500393922573853964274194191, −0.977655826003258989203126608043,
1.30787914276028695787481620968, 2.59387069094861691943692945060, 3.15580694498699688907818074255, 4.15406023457178777817247351057, 4.91174045594485459591288673913, 6.23378615122407586038169321184, 6.78333224310095492196369914450, 7.67022962994719693340112156167, 8.350229568311416250194602215632, 9.062291615429677688826939035424