| L(s) = 1 | − 2-s − 3-s + 4-s + 0.356·5-s + 6-s + 7-s − 8-s + 9-s − 0.356·10-s − 3.40·11-s − 12-s − 14-s − 0.356·15-s + 16-s − 2.66·17-s − 18-s − 8.63·19-s + 0.356·20-s − 21-s + 3.40·22-s + 8.29·23-s + 24-s − 4.87·25-s − 27-s + 28-s + 2.21·29-s + 0.356·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.159·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.112·10-s − 1.02·11-s − 0.288·12-s − 0.267·14-s − 0.0921·15-s + 0.250·16-s − 0.646·17-s − 0.235·18-s − 1.98·19-s + 0.0798·20-s − 0.218·21-s + 0.726·22-s + 1.72·23-s + 0.204·24-s − 0.974·25-s − 0.192·27-s + 0.188·28-s + 0.412·29-s + 0.0651·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.356T + 5T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 17 | \( 1 + 2.66T + 17T^{2} \) |
| 19 | \( 1 + 8.63T + 19T^{2} \) |
| 23 | \( 1 - 8.29T + 23T^{2} \) |
| 29 | \( 1 - 2.21T + 29T^{2} \) |
| 31 | \( 1 - 3.89T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 - 6.66T + 41T^{2} \) |
| 43 | \( 1 - 5.82T + 43T^{2} \) |
| 47 | \( 1 + 4.59T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 3.78T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 1.72T + 73T^{2} \) |
| 79 | \( 1 - 4.49T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 9.28T + 97T^{2} \) |
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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
−7.75694781236027676084112750591, −6.87397414212527089686602359988, −6.26204329766499321445636992326, −5.64742534427441715404395395920, −4.69609791820562719735919245814, −4.21384417937094046376613281758, −2.77434688956749138652460658513, −2.24329291388236527287887303178, −1.08806988962437790297298111951, 0,
1.08806988962437790297298111951, 2.24329291388236527287887303178, 2.77434688956749138652460658513, 4.21384417937094046376613281758, 4.69609791820562719735919245814, 5.64742534427441715404395395920, 6.26204329766499321445636992326, 6.87397414212527089686602359988, 7.75694781236027676084112750591
