| L(s) = 1 | − 3-s + 4·7-s + 9-s + 2·11-s − 2·13-s − 6·19-s − 4·21-s − 27-s + 10·29-s − 8·31-s − 2·33-s + 4·37-s + 2·39-s − 10·41-s − 43-s + 9·49-s − 12·53-s + 6·57-s + 6·59-s − 10·61-s + 4·63-s + 12·67-s − 4·71-s + 8·73-s + 8·77-s + 16·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 1.37·19-s − 0.872·21-s − 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.348·33-s + 0.657·37-s + 0.320·39-s − 1.56·41-s − 0.152·43-s + 9/7·49-s − 1.64·53-s + 0.794·57-s + 0.781·59-s − 1.28·61-s + 0.503·63-s + 1.46·67-s − 0.474·71-s + 0.936·73-s + 0.911·77-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.962935671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.962935671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.49468267765332, −14.08810030997948, −13.56638420433977, −12.73220055250599, −12.41150912697524, −11.91425313595242, −11.30088961171928, −11.02974727621944, −10.48943427711687, −9.920181674909167, −9.296721773549003, −8.609378731861663, −8.206244166175732, −7.741472359624609, −6.904114616203141, −6.616807181477894, −5.910543430917178, −5.190285266964913, −4.770500590606104, −4.334453162074023, −3.647941009017879, −2.695918432550934, −1.890860050634127, −1.483084623967288, −0.5124688515613306,
0.5124688515613306, 1.483084623967288, 1.890860050634127, 2.695918432550934, 3.647941009017879, 4.334453162074023, 4.770500590606104, 5.190285266964913, 5.910543430917178, 6.616807181477894, 6.904114616203141, 7.741472359624609, 8.206244166175732, 8.609378731861663, 9.296721773549003, 9.920181674909167, 10.48943427711687, 11.02974727621944, 11.30088961171928, 11.91425313595242, 12.41150912697524, 12.73220055250599, 13.56638420433977, 14.08810030997948, 14.49468267765332
