L(s) = 1 | + (−1.41 − i)3-s − 2.82·5-s − 2.82i·7-s + (1.00 + 2.82i)9-s + 2i·11-s + 4i·13-s + (4.00 + 2.82i)15-s + 5.65i·17-s + 2.82·19-s + (−2.82 + 4.00i)21-s − 8·23-s + 3.00·25-s + (1.41 − 5.00i)27-s − 2.82·29-s + 8.48i·31-s + ⋯ |
L(s) = 1 | + (−0.816 − 0.577i)3-s − 1.26·5-s − 1.06i·7-s + (0.333 + 0.942i)9-s + 0.603i·11-s + 1.10i·13-s + (1.03 + 0.730i)15-s + 1.37i·17-s + 0.648·19-s + (−0.617 + 0.872i)21-s − 1.66·23-s + 0.600·25-s + (0.272 − 0.962i)27-s − 0.525·29-s + 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.228262 + 0.270762i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.228262 + 0.270762i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 + i)T \) |
good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 5.65iT - 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 4iT - 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−11.84013431849543487624085734816, −10.75324678714745173550218605807, −10.13865646447775057252542968485, −8.580562330235103347671431313467, −7.51671614009728113911271684319, −7.13901766699093494263261841770, −5.97878864229280295301841996051, −4.48105143546737670939677393280, −3.86827606007007513778805288423, −1.59836086660238128205971314312,
0.26711616371200848239957197239, 3.00006713872455957799728131876, 4.09543316838642553122278100661, 5.30570611399189772799361049528, 5.99830149779478165528715613354, 7.42819174845316496691319549424, 8.265909017699831400638512889083, 9.350172957186352377738380075439, 10.23765218493591380398194250325, 11.49946274052800287393322569326