| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 13-s + 15-s − 17-s − 4·19-s + 2·21-s + 25-s − 27-s − 6·29-s − 10·31-s + 2·35-s + 4·37-s + 39-s − 2·41-s − 4·43-s − 45-s − 6·47-s − 3·49-s + 51-s + 6·53-s + 4·57-s + 6·59-s − 6·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.917·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.79·31-s + 0.338·35-s + 0.657·37-s + 0.160·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.140·51-s + 0.824·53-s + 0.529·57-s + 0.781·59-s − 0.768·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−13.01046464976294, −12.75808801751058, −12.50608119140027, −11.69618699073246, −11.32848524561632, −11.11750867808000, −10.35796576899188, −10.07991054514978, −9.514790944648676, −9.020265745836946, −8.599443504256487, −7.940206332612462, −7.438885908415779, −7.010521010585127, −6.490513211020171, −6.123941974382743, −5.387687649821239, −5.143586235503054, −4.317236848609114, −3.940535932614739, −3.430122155360985, −2.790107728297388, −2.042569546712105, −1.543929070796321, −0.5060396014699739, 0,
0.5060396014699739, 1.543929070796321, 2.042569546712105, 2.790107728297388, 3.430122155360985, 3.940535932614739, 4.317236848609114, 5.143586235503054, 5.387687649821239, 6.123941974382743, 6.490513211020171, 7.010521010585127, 7.438885908415779, 7.940206332612462, 8.599443504256487, 9.020265745836946, 9.514790944648676, 10.07991054514978, 10.35796576899188, 11.11750867808000, 11.32848524561632, 11.69618699073246, 12.50608119140027, 12.75808801751058, 13.01046464976294
