L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 + 0.707i)8-s + (−0.707 − 1.22i)11-s + (0.500 − 0.866i)16-s + (−0.999 + i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s − 1.41i·29-s + (−0.965 − 0.258i)32-s − 2i·43-s + (1.22 + 0.707i)44-s + (−1.36 − 0.366i)46-s + (0.707 − 0.707i)50-s + (1.22 − 0.707i)53-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 + 0.707i)8-s + (−0.707 − 1.22i)11-s + (0.500 − 0.866i)16-s + (−0.999 + i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s − 1.41i·29-s + (−0.965 − 0.258i)32-s − 2i·43-s + (1.22 + 0.707i)44-s + (−1.36 − 0.366i)46-s + (0.707 − 0.707i)50-s + (1.22 − 0.707i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8006908255\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8006908255\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 2iT - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.157860816336504019867756337743, −8.626559676456714203044461334188, −7.937485561176276576174946025553, −6.98705591903434316981579439418, −5.79035220734050329317034061449, −5.03695861312447982139042669599, −3.99803231242290485220090124082, −3.09598369952669118477319286680, −2.25810675944303055153292687532, −0.71602828464061055989823116064,
1.46894219978427030265296706435, 2.95818914581082294480088945019, 4.29430883044022573514995314443, 4.97859036284355183943643589591, 5.73841459856037474768431432644, 6.79174643681991107499800273136, 7.32942424870204003464296546637, 8.045092299958424697628204092418, 8.919371237152498304896583209779, 9.602319398826738634569022836515