| L(s) = 1 | + 7-s − 3·9-s − 2·11-s − 2·13-s − 4·17-s + 19-s + 9·23-s − 5·25-s − 2·29-s + 2·31-s − 8·37-s − 6·41-s + 9·43-s + 4·47-s − 6·49-s − 3·63-s − 2·67-s + 16·71-s + 73-s − 2·77-s + 5·79-s + 9·81-s + 9·83-s + 10·89-s − 2·91-s + 12·97-s + 6·99-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 0.603·11-s − 0.554·13-s − 0.970·17-s + 0.229·19-s + 1.87·23-s − 25-s − 0.371·29-s + 0.359·31-s − 1.31·37-s − 0.937·41-s + 1.37·43-s + 0.583·47-s − 6/7·49-s − 0.377·63-s − 0.244·67-s + 1.89·71-s + 0.117·73-s − 0.227·77-s + 0.562·79-s + 81-s + 0.987·83-s + 1.05·89-s − 0.209·91-s + 1.21·97-s + 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 73 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.53508868955856, −13.20669575892824, −12.35700836370522, −12.24525611250089, −11.47789536340151, −11.18323799182487, −10.71646289617631, −10.35786518711316, −9.473497592515646, −9.267407772460543, −8.715723100639199, −8.194557983789793, −7.764217108129125, −7.220256036985226, −6.671771420053137, −6.215451434682881, −5.364654766686913, −5.207889482616347, −4.730812505530552, −3.919025179633977, −3.388140995597105, −2.686449391049968, −2.333491365984522, −1.621362739836752, −0.6981127782081277, 0,
0.6981127782081277, 1.621362739836752, 2.333491365984522, 2.686449391049968, 3.388140995597105, 3.919025179633977, 4.730812505530552, 5.207889482616347, 5.364654766686913, 6.215451434682881, 6.671771420053137, 7.220256036985226, 7.764217108129125, 8.194557983789793, 8.715723100639199, 9.267407772460543, 9.473497592515646, 10.35786518711316, 10.71646289617631, 11.18323799182487, 11.47789536340151, 12.24525611250089, 12.35700836370522, 13.20669575892824, 13.53508868955856
