| L(s) = 1 | + 5·5-s + 13·7-s − 30·11-s − 61·13-s − 12·17-s + 49·19-s + 18·23-s + 25·25-s + 186·29-s + 160·31-s + 65·35-s − 91·37-s − 378·41-s + 268·43-s + 144·47-s − 174·49-s − 570·53-s − 150·55-s + 204·59-s − 877·61-s − 305·65-s + 187·67-s − 606·71-s + 431·73-s − 390·77-s − 1.15e3·79-s + 102·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.701·7-s − 0.822·11-s − 1.30·13-s − 0.171·17-s + 0.591·19-s + 0.163·23-s + 1/5·25-s + 1.19·29-s + 0.926·31-s + 0.313·35-s − 0.404·37-s − 1.43·41-s + 0.950·43-s + 0.446·47-s − 0.507·49-s − 1.47·53-s − 0.367·55-s + 0.450·59-s − 1.84·61-s − 0.582·65-s + 0.340·67-s − 1.01·71-s + 0.691·73-s − 0.577·77-s − 1.63·79-s + 0.134·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 13 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 61 T + p^{3} T^{2} \) |
| 17 | \( 1 + 12 T + p^{3} T^{2} \) |
| 19 | \( 1 - 49 T + p^{3} T^{2} \) |
| 23 | \( 1 - 18 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 - 160 T + p^{3} T^{2} \) |
| 37 | \( 1 + 91 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 268 T + p^{3} T^{2} \) |
| 47 | \( 1 - 144 T + p^{3} T^{2} \) |
| 53 | \( 1 + 570 T + p^{3} T^{2} \) |
| 59 | \( 1 - 204 T + p^{3} T^{2} \) |
| 61 | \( 1 + 877 T + p^{3} T^{2} \) |
| 67 | \( 1 - 187 T + p^{3} T^{2} \) |
| 71 | \( 1 + 606 T + p^{3} T^{2} \) |
| 73 | \( 1 - 431 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1151 T + p^{3} T^{2} \) |
| 83 | \( 1 - 102 T + p^{3} T^{2} \) |
| 89 | \( 1 + 984 T + p^{3} T^{2} \) |
| 97 | \( 1 + 265 T + p^{3} T^{2} \) |
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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
−8.254736315111238226239990837141, −7.61758088739838541673693953830, −6.85494053936086481919441013014, −5.88715128711167053414543774297, −4.95145852766547221958457546677, −4.63598256605732546351459510792, −3.11121510301204694989584467281, −2.38468134220368854215018624376, −1.32401978956667316935410870757, 0,
1.32401978956667316935410870757, 2.38468134220368854215018624376, 3.11121510301204694989584467281, 4.63598256605732546351459510792, 4.95145852766547221958457546677, 5.88715128711167053414543774297, 6.85494053936086481919441013014, 7.61758088739838541673693953830, 8.254736315111238226239990837141
