| L(s) = 1 | − 3-s − 3.85·5-s + 0.236·7-s + 9-s + 6.23·13-s + 3.85·15-s − 2.38·17-s + 5.09·19-s − 0.236·21-s − 0.236·23-s + 9.85·25-s − 27-s − 4.47·29-s − 6.38·31-s − 0.909·35-s − 3.76·37-s − 6.23·39-s − 9.47·41-s − 9.47·43-s − 3.85·45-s + 3.61·47-s − 6.94·49-s + 2.38·51-s + 6.56·53-s − 5.09·57-s + 2.14·59-s + 2.61·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.72·5-s + 0.0892·7-s + 0.333·9-s + 1.72·13-s + 0.995·15-s − 0.577·17-s + 1.16·19-s − 0.0515·21-s − 0.0492·23-s + 1.97·25-s − 0.192·27-s − 0.830·29-s − 1.14·31-s − 0.153·35-s − 0.618·37-s − 0.998·39-s − 1.47·41-s − 1.44·43-s − 0.574·45-s + 0.527·47-s − 0.992·49-s + 0.333·51-s + 0.901·53-s − 0.674·57-s + 0.279·59-s + 0.335·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 - 5.09T + 19T^{2} \) |
| 23 | \( 1 + 0.236T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.38T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 + 9.47T + 41T^{2} \) |
| 43 | \( 1 + 9.47T + 43T^{2} \) |
| 47 | \( 1 - 3.61T + 47T^{2} \) |
| 53 | \( 1 - 6.56T + 53T^{2} \) |
| 59 | \( 1 - 2.14T + 59T^{2} \) |
| 61 | \( 1 - 2.61T + 61T^{2} \) |
| 67 | \( 1 + 0.145T + 67T^{2} \) |
| 71 | \( 1 + 1.38T + 71T^{2} \) |
| 73 | \( 1 + 3.23T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + 5.09T + 97T^{2} \) |
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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
−8.830341188319842416553234282663, −8.380270746556061855414968287281, −7.40973026357062656699246483417, −6.86299248002417109606322716945, −5.76966136436514785605235865344, −4.86290527847168790090088235763, −3.83075727509495184414864371535, −3.38439668111259848109782356332, −1.41224756078092261457138151205, 0,
1.41224756078092261457138151205, 3.38439668111259848109782356332, 3.83075727509495184414864371535, 4.86290527847168790090088235763, 5.76966136436514785605235865344, 6.86299248002417109606322716945, 7.40973026357062656699246483417, 8.380270746556061855414968287281, 8.830341188319842416553234282663
