L(s) = 1 | + (−1.70 + 1.23i)2-s + (0.749 − 2.30i)4-s + (−3.03 − 2.20i)5-s + (0.309 − 0.951i)7-s + (0.277 + 0.853i)8-s + 7.89·10-s + (−1.33 − 3.03i)11-s + (−2.80 + 2.03i)13-s + (0.650 + 2.00i)14-s + (2.39 + 1.74i)16-s + (1.23 + 0.897i)17-s + (−1.59 − 4.90i)19-s + (−7.36 + 5.34i)20-s + (6.02 + 3.51i)22-s + 4.87·23-s + ⋯ |
L(s) = 1 | + (−1.20 + 0.874i)2-s + (0.374 − 1.15i)4-s + (−1.35 − 0.985i)5-s + (0.116 − 0.359i)7-s + (0.0981 + 0.301i)8-s + 2.49·10-s + (−0.402 − 0.915i)11-s + (−0.777 + 0.564i)13-s + (0.173 + 0.534i)14-s + (0.599 + 0.435i)16-s + (0.299 + 0.217i)17-s + (−0.365 − 1.12i)19-s + (−1.64 + 1.19i)20-s + (1.28 + 0.749i)22-s + 1.01·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0402618 + 0.142235i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0402618 + 0.142235i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (1.33 + 3.03i)T \) |
good | 2 | \( 1 + (1.70 - 1.23i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (3.03 + 2.20i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (2.80 - 2.03i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.23 - 0.897i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.59 + 4.90i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 4.87T + 23T^{2} \) |
| 29 | \( 1 + (3.31 - 10.2i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (6.57 - 4.77i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.86 + 8.80i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.36 - 10.3i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 0.137T + 43T^{2} \) |
| 47 | \( 1 + (-0.843 - 2.59i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2.26 - 1.64i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.49 + 4.60i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.240 - 0.175i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 0.816T + 67T^{2} \) |
| 71 | \( 1 + (-4.63 - 3.36i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.764 - 2.35i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (9.24 - 6.71i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (8.61 + 6.25i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 8.91T + 89T^{2} \) |
| 97 | \( 1 + (6.54 - 4.75i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.92176503365162735922016843741, −9.487613036644731990083544432430, −8.863437305191443149021925984421, −8.312442636735994246557200533458, −7.38248987065687947018989500433, −6.99669600938470729419606765594, −5.49957294497338536283389218999, −4.56434813685770590707814349365, −3.36516350434482609143538140919, −1.06239692231568966167010832102,
0.14097979799502915680738338729, 2.15991436660307451780936482144, 3.05216027022101446566129517524, 4.18208566550631467266044508526, 5.60435917221604076992119143718, 7.17004612842868107246734126116, 7.69899978522359774790417300407, 8.299869102733182240638416180841, 9.517426944274575037279706515156, 10.14483981586047557055992875520