L(s) = 1 | + 12·5-s + 7·7-s − 60·11-s − 79·13-s − 108·17-s − 11·19-s + 132·23-s + 19·25-s + 96·29-s − 20·31-s + 84·35-s − 169·37-s + 192·41-s − 488·43-s − 204·47-s − 294·49-s + 360·53-s − 720·55-s − 156·59-s + 83·61-s − 948·65-s − 47·67-s − 216·71-s − 511·73-s − 420·77-s + 529·79-s + 1.12e3·83-s + ⋯ |
L(s) = 1 | + 1.07·5-s + 0.377·7-s − 1.64·11-s − 1.68·13-s − 1.54·17-s − 0.132·19-s + 1.19·23-s + 0.151·25-s + 0.614·29-s − 0.115·31-s + 0.405·35-s − 0.750·37-s + 0.731·41-s − 1.73·43-s − 0.633·47-s − 6/7·49-s + 0.933·53-s − 1.76·55-s − 0.344·59-s + 0.174·61-s − 1.80·65-s − 0.0857·67-s − 0.361·71-s − 0.819·73-s − 0.621·77-s + 0.753·79-s + 1.49·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 7 | \( 1 - p T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 79 T + p^{3} T^{2} \) |
| 17 | \( 1 + 108 T + p^{3} T^{2} \) |
| 19 | \( 1 + 11 T + p^{3} T^{2} \) |
| 23 | \( 1 - 132 T + p^{3} T^{2} \) |
| 29 | \( 1 - 96 T + p^{3} T^{2} \) |
| 31 | \( 1 + 20 T + p^{3} T^{2} \) |
| 37 | \( 1 + 169 T + p^{3} T^{2} \) |
| 41 | \( 1 - 192 T + p^{3} T^{2} \) |
| 43 | \( 1 + 488 T + p^{3} T^{2} \) |
| 47 | \( 1 + 204 T + p^{3} T^{2} \) |
| 53 | \( 1 - 360 T + p^{3} T^{2} \) |
| 59 | \( 1 + 156 T + p^{3} T^{2} \) |
| 61 | \( 1 - 83 T + p^{3} T^{2} \) |
| 67 | \( 1 + 47 T + p^{3} T^{2} \) |
| 71 | \( 1 + 216 T + p^{3} T^{2} \) |
| 73 | \( 1 + 7 p T + p^{3} T^{2} \) |
| 79 | \( 1 - 529 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1128 T + p^{3} T^{2} \) |
| 89 | \( 1 - 36 T + p^{3} T^{2} \) |
| 97 | \( 1 - 605 T + p^{3} T^{2} \) |
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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
−10.27198730293778426338633718512, −9.508905254275148629459823395688, −8.505381719761003929894200142230, −7.46024021178833728381368460831, −6.54158592585201217943873426615, −5.22086457610393949822192305394, −4.80267990136186611383193490329, −2.77088996863027451313297791694, −2.01353307260586976014018082412, 0,
2.01353307260586976014018082412, 2.77088996863027451313297791694, 4.80267990136186611383193490329, 5.22086457610393949822192305394, 6.54158592585201217943873426615, 7.46024021178833728381368460831, 8.505381719761003929894200142230, 9.508905254275148629459823395688, 10.27198730293778426338633718512