| L(s) = 1 | − 1.24·2-s − 3-s − 0.445·4-s − 4.06·5-s + 1.24·6-s + 7-s + 3.04·8-s + 9-s + 5.07·10-s + 0.456·11-s + 0.445·12-s − 1.24·14-s + 4.06·15-s − 2.91·16-s − 5.72·17-s − 1.24·18-s − 0.188·19-s + 1.81·20-s − 21-s − 0.569·22-s − 5.05·23-s − 3.04·24-s + 11.5·25-s − 27-s − 0.445·28-s − 6.16·29-s − 5.07·30-s + ⋯ |
| L(s) = 1 | − 0.881·2-s − 0.577·3-s − 0.222·4-s − 1.82·5-s + 0.509·6-s + 0.377·7-s + 1.07·8-s + 0.333·9-s + 1.60·10-s + 0.137·11-s + 0.128·12-s − 0.333·14-s + 1.05·15-s − 0.727·16-s − 1.38·17-s − 0.293·18-s − 0.0432·19-s + 0.405·20-s − 0.218·21-s − 0.121·22-s − 1.05·23-s − 0.622·24-s + 2.31·25-s − 0.192·27-s − 0.0841·28-s − 1.14·29-s − 0.926·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 + 4.06T + 5T^{2} \) |
| 11 | \( 1 - 0.456T + 11T^{2} \) |
| 17 | \( 1 + 5.72T + 17T^{2} \) |
| 19 | \( 1 + 0.188T + 19T^{2} \) |
| 23 | \( 1 + 5.05T + 23T^{2} \) |
| 29 | \( 1 + 6.16T + 29T^{2} \) |
| 31 | \( 1 + 1.19T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 2.45T + 41T^{2} \) |
| 43 | \( 1 - 4.28T + 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 0.497T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 7.94T + 73T^{2} \) |
| 79 | \( 1 - 4.07T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 9.85T + 97T^{2} \) |
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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
−8.217382649420087821138492112096, −7.51729276962334127006265426469, −7.15540648539554891482100639589, −6.05010029552143334038589746260, −4.95489918998708656287555295067, −4.13943503523424813278644619811, −3.95428697814206476871347516653, −2.28565680993371163228877761043, −0.886992533021107097017140884378, 0,
0.886992533021107097017140884378, 2.28565680993371163228877761043, 3.95428697814206476871347516653, 4.13943503523424813278644619811, 4.95489918998708656287555295067, 6.05010029552143334038589746260, 7.15540648539554891482100639589, 7.51729276962334127006265426469, 8.217382649420087821138492112096
