L(s) = 1 | + i·3-s + (−1 − 2i)5-s + i·7-s − 9-s + 2·11-s + 2i·13-s + (2 − i)15-s + 6·19-s − 21-s + (−3 + 4i)25-s − i·27-s + 6·29-s + 10·31-s + 2i·33-s + (2 − i)35-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.447 − 0.894i)5-s + 0.377i·7-s − 0.333·9-s + 0.603·11-s + 0.554i·13-s + (0.516 − 0.258i)15-s + 1.37·19-s − 0.218·21-s + (−0.600 + 0.800i)25-s − 0.192i·27-s + 1.11·29-s + 1.79·31-s + 0.348i·33-s + (0.338 − 0.169i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45247 + 0.342882i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45247 + 0.342882i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−10.02695330967953646866738060766, −9.418011676044524388548164316249, −8.663249985612152757254746691212, −7.943842594689651466213394460428, −6.78580328359841470632783419403, −5.71681709487707030678622672176, −4.78787157450412168484313680364, −4.07404136542327940724636991805, −2.87066147237433745763767703463, −1.14211722010438502308014106984,
0.968338796837998335226964977087, 2.65807631367547533223490342958, 3.51748428969480125427180431368, 4.71959877830515934641097957805, 6.04024853233344267166078629801, 6.73412513690147874693435498117, 7.58537639148432645329750212349, 8.152879485491472851596332374530, 9.357583811269004123427876476300, 10.24622376850162292838146886493