| L(s) = 1 | + 60·3-s − 2.07e3·5-s + 4.34e3·7-s − 1.60e4·9-s − 9.36e4·11-s − 1.22e4·13-s − 1.24e5·15-s − 3.19e5·17-s + 5.53e5·19-s + 2.60e5·21-s + 7.12e5·23-s + 2.34e6·25-s − 2.14e6·27-s + 2.07e6·29-s + 6.42e6·31-s − 5.61e6·33-s − 9.00e6·35-s − 1.81e7·37-s − 7.34e5·39-s + 9.03e6·41-s − 1.95e7·43-s + 3.33e7·45-s + 1.84e7·47-s − 2.14e7·49-s − 1.91e7·51-s + 1.02e7·53-s + 1.94e8·55-s + ⋯ |
| L(s) = 1 | + 0.427·3-s − 1.48·5-s + 0.683·7-s − 0.817·9-s − 1.92·11-s − 0.118·13-s − 0.634·15-s − 0.928·17-s + 0.974·19-s + 0.292·21-s + 0.531·23-s + 1.20·25-s − 0.777·27-s + 0.545·29-s + 1.24·31-s − 0.824·33-s − 1.01·35-s − 1.59·37-s − 0.0508·39-s + 0.499·41-s − 0.874·43-s + 1.21·45-s + 0.552·47-s − 0.532·49-s − 0.396·51-s + 0.178·53-s + 2.86·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 20 p T + p^{9} T^{2} \) |
| 5 | \( 1 + 2074 T + p^{9} T^{2} \) |
| 7 | \( 1 - 4344 T + p^{9} T^{2} \) |
| 11 | \( 1 + 93644 T + p^{9} T^{2} \) |
| 13 | \( 1 + 12242 T + p^{9} T^{2} \) |
| 17 | \( 1 + 319598 T + p^{9} T^{2} \) |
| 19 | \( 1 - 553516 T + p^{9} T^{2} \) |
| 23 | \( 1 - 712936 T + p^{9} T^{2} \) |
| 29 | \( 1 - 2075838 T + p^{9} T^{2} \) |
| 31 | \( 1 - 6420448 T + p^{9} T^{2} \) |
| 37 | \( 1 + 18197754 T + p^{9} T^{2} \) |
| 41 | \( 1 - 9033834 T + p^{9} T^{2} \) |
| 43 | \( 1 + 19594732 T + p^{9} T^{2} \) |
| 47 | \( 1 - 18484176 T + p^{9} T^{2} \) |
| 53 | \( 1 - 10255766 T + p^{9} T^{2} \) |
| 59 | \( 1 + 121666556 T + p^{9} T^{2} \) |
| 61 | \( 1 + 45948962 T + p^{9} T^{2} \) |
| 67 | \( 1 + 50535428 T + p^{9} T^{2} \) |
| 71 | \( 1 + 267044680 T + p^{9} T^{2} \) |
| 73 | \( 1 + 176213366 T + p^{9} T^{2} \) |
| 79 | \( 1 - 269685680 T + p^{9} T^{2} \) |
| 83 | \( 1 - 2735332 p T + p^{9} T^{2} \) |
| 89 | \( 1 - 72141594 T + p^{9} T^{2} \) |
| 97 | \( 1 - 228776546 T + p^{9} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.91325768747657790604796197562, −15.15515921892434168010546645610, −13.62723336403881993101610562146, −11.93363310227415215810803553645, −10.79710314026649805254609511508, −8.481114066913156409080098223892, −7.61396248802633615613961716723, −4.91231409850642751106607172873, −2.93783219260524971328797366809, 0,
2.93783219260524971328797366809, 4.91231409850642751106607172873, 7.61396248802633615613961716723, 8.481114066913156409080098223892, 10.79710314026649805254609511508, 11.93363310227415215810803553645, 13.62723336403881993101610562146, 15.15515921892434168010546645610, 15.91325768747657790604796197562
