| L(s) = 1 | + 3-s + 9-s + 4·11-s − 2·13-s − 17-s − 4·19-s + 27-s + 10·29-s + 8·31-s + 4·33-s − 2·37-s − 2·39-s + 10·41-s + 12·43-s − 7·49-s − 51-s + 6·53-s − 4·57-s − 12·59-s + 10·61-s − 12·67-s − 10·73-s − 8·79-s + 81-s + 4·83-s + 10·87-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.242·17-s − 0.917·19-s + 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.696·33-s − 0.328·37-s − 0.320·39-s + 1.56·41-s + 1.82·43-s − 49-s − 0.140·51-s + 0.824·53-s − 0.529·57-s − 1.56·59-s + 1.28·61-s − 1.46·67-s − 1.17·73-s − 0.900·79-s + 1/9·81-s + 0.439·83-s + 1.07·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.743321964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.743321964\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.21202329657305, −13.53870668830553, −12.99416366332152, −12.45474342631961, −12.05167529814721, −11.61245840395449, −10.94079048807223, −10.37321051742247, −9.996658682751705, −9.346615981088966, −8.912745524270190, −8.518836134825939, −7.933382126851542, −7.323398875989301, −6.852540570175349, −6.156496768439250, −6.001842692009561, −4.842571293776780, −4.413253748885287, −4.124490117962280, −3.204348345407268, −2.693373126494427, −2.121992942409625, −1.280543811639999, −0.6455115451686259,
0.6455115451686259, 1.280543811639999, 2.121992942409625, 2.693373126494427, 3.204348345407268, 4.124490117962280, 4.413253748885287, 4.842571293776780, 6.001842692009561, 6.156496768439250, 6.852540570175349, 7.323398875989301, 7.933382126851542, 8.518836134825939, 8.912745524270190, 9.346615981088966, 9.996658682751705, 10.37321051742247, 10.94079048807223, 11.61245840395449, 12.05167529814721, 12.45474342631961, 12.99416366332152, 13.53870668830553, 14.21202329657305
