| L(s) = 1 | − 2-s + 0.554·3-s + 4-s − 1.35·5-s − 0.554·6-s + 7-s − 8-s − 2.69·9-s + 1.35·10-s − 1.80·11-s + 0.554·12-s − 14-s − 0.753·15-s + 16-s + 5.29·17-s + 2.69·18-s − 1.95·19-s − 1.35·20-s + 0.554·21-s + 1.80·22-s + 4.85·23-s − 0.554·24-s − 3.15·25-s − 3.15·27-s + 28-s + 5.96·29-s + 0.753·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.320·3-s + 0.5·4-s − 0.606·5-s − 0.226·6-s + 0.377·7-s − 0.353·8-s − 0.897·9-s + 0.429·10-s − 0.543·11-s + 0.160·12-s − 0.267·14-s − 0.194·15-s + 0.250·16-s + 1.28·17-s + 0.634·18-s − 0.447·19-s − 0.303·20-s + 0.121·21-s + 0.384·22-s + 1.01·23-s − 0.113·24-s − 0.631·25-s − 0.607·27-s + 0.188·28-s + 1.10·29-s + 0.137·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.554T + 3T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 29 | \( 1 - 5.96T + 29T^{2} \) |
| 31 | \( 1 - 3.63T + 31T^{2} \) |
| 37 | \( 1 + 7.75T + 37T^{2} \) |
| 41 | \( 1 + 0.0392T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 3.66T + 53T^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 2.22T + 67T^{2} \) |
| 71 | \( 1 + 7.78T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 2.85T + 83T^{2} \) |
| 89 | \( 1 - 6.61T + 89T^{2} \) |
| 97 | \( 1 - 3.64T + 97T^{2} \) |
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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
−8.485390853737094288755200888993, −7.968155363061284597706037577787, −7.39838861335867833538584109743, −6.36757103317691980418833322569, −5.50002064250011894038264864989, −4.61944219819896122327841880857, −3.35216503007983102113929554716, −2.76281468836272425846153540782, −1.43798903495050729391790759829, 0,
1.43798903495050729391790759829, 2.76281468836272425846153540782, 3.35216503007983102113929554716, 4.61944219819896122327841880857, 5.50002064250011894038264864989, 6.36757103317691980418833322569, 7.39838861335867833538584109743, 7.968155363061284597706037577787, 8.485390853737094288755200888993
