| L(s) = 1 | − 2·2-s + 3-s + 4·4-s − 7·5-s − 2·6-s − 8·8-s − 26·9-s + 14·10-s + 35·11-s + 4·12-s − 66·13-s − 7·15-s + 16·16-s − 59·17-s + 52·18-s − 137·19-s − 28·20-s − 70·22-s − 7·23-s − 8·24-s − 76·25-s + 132·26-s − 53·27-s + 106·29-s + 14·30-s − 75·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.192·3-s + 1/2·4-s − 0.626·5-s − 0.136·6-s − 0.353·8-s − 0.962·9-s + 0.442·10-s + 0.959·11-s + 0.0962·12-s − 1.40·13-s − 0.120·15-s + 1/4·16-s − 0.841·17-s + 0.680·18-s − 1.65·19-s − 0.313·20-s − 0.678·22-s − 0.0634·23-s − 0.0680·24-s − 0.607·25-s + 0.995·26-s − 0.377·27-s + 0.678·29-s + 0.0852·30-s − 0.434·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 - 35 T + p^{3} T^{2} \) |
| 13 | \( 1 + 66 T + p^{3} T^{2} \) |
| 17 | \( 1 + 59 T + p^{3} T^{2} \) |
| 19 | \( 1 + 137 T + p^{3} T^{2} \) |
| 23 | \( 1 + 7 T + p^{3} T^{2} \) |
| 29 | \( 1 - 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 75 T + p^{3} T^{2} \) |
| 37 | \( 1 - 11 T + p^{3} T^{2} \) |
| 41 | \( 1 - 498 T + p^{3} T^{2} \) |
| 43 | \( 1 - 260 T + p^{3} T^{2} \) |
| 47 | \( 1 - 171 T + p^{3} T^{2} \) |
| 53 | \( 1 + 417 T + p^{3} T^{2} \) |
| 59 | \( 1 - 17 T + p^{3} T^{2} \) |
| 61 | \( 1 + 51 T + p^{3} T^{2} \) |
| 67 | \( 1 - 439 T + p^{3} T^{2} \) |
| 71 | \( 1 + 784 T + p^{3} T^{2} \) |
| 73 | \( 1 + 295 T + p^{3} T^{2} \) |
| 79 | \( 1 + 495 T + p^{3} T^{2} \) |
| 83 | \( 1 + 932 T + p^{3} T^{2} \) |
| 89 | \( 1 - 873 T + p^{3} T^{2} \) |
| 97 | \( 1 - 290 T + p^{3} T^{2} \) |
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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
−12.63011147720744874618543591407, −11.69918690882632076491328556884, −10.80401810264659518389797641726, −9.409823262858890820269855256226, −8.552560075385763661490924182139, −7.43753863824798345846939812513, −6.18398972845642701149808383127, −4.26518710107502167053907221998, −2.43739548059118331864103691866, 0,
2.43739548059118331864103691866, 4.26518710107502167053907221998, 6.18398972845642701149808383127, 7.43753863824798345846939812513, 8.552560075385763661490924182139, 9.409823262858890820269855256226, 10.80401810264659518389797641726, 11.69918690882632076491328556884, 12.63011147720744874618543591407
