L(s) = 1 | + 2.80i·2-s − 5.87·4-s − 2.23i·5-s − 10.8i·8-s + 6.27·10-s + 18.7·16-s + 6.92i·17-s + 2.64·19-s + 13.1i·20-s − 3.87i·23-s − 5.00·25-s − 11.1·31-s + 30.8i·32-s − 19.4·34-s + 7.43i·38-s + ⋯ |
L(s) = 1 | + 1.98i·2-s − 2.93·4-s − 0.999i·5-s − 3.84i·8-s + 1.98·10-s + 4.68·16-s + 1.68i·17-s + 0.607·19-s + 2.93i·20-s − 0.808i·23-s − 1.00·25-s − 1.99·31-s + 5.45i·32-s − 3.33·34-s + 1.20i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2338952134\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2338952134\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.80iT - 2T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6.92iT - 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 + 3.87iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 11.1T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 6.63iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 7.74T + 79T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.178503027106339908659706631611, −8.667686832760434703838194778009, −7.983852899200033291614137158315, −7.42147131648609445745934474918, −6.40840323681308140300457560335, −5.81290395532023858514164258948, −5.12743011198655494285084663518, −4.32285393716670954541893114606, −3.63009462081194942617221027632, −1.36350268846119497953263046102,
0.088865032710439574633104844232, 1.55531302515044527648083553886, 2.56808002917694823314147076690, 3.23657021131622238725684049056, 3.94455606513835219534204681204, 5.04358442157105211849228982314, 5.69021360371806604437040450048, 7.17377430509722040432963476778, 7.81132868592604727598025106471, 9.047079238402902757003473367812