| L(s) = 1 | + 3·3-s − 4·5-s + 7-s + 6·9-s + 2·11-s − 5·13-s − 12·15-s − 4·19-s + 3·21-s + 11·25-s + 9·27-s + 3·29-s + 5·31-s + 6·33-s − 4·35-s + 4·37-s − 15·39-s + 5·41-s − 4·43-s − 24·45-s − 11·47-s + 49-s − 8·55-s − 12·57-s + 12·59-s − 6·61-s + 6·63-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.78·5-s + 0.377·7-s + 2·9-s + 0.603·11-s − 1.38·13-s − 3.09·15-s − 0.917·19-s + 0.654·21-s + 11/5·25-s + 1.73·27-s + 0.557·29-s + 0.898·31-s + 1.04·33-s − 0.676·35-s + 0.657·37-s − 2.40·39-s + 0.780·41-s − 0.609·43-s − 3.57·45-s − 1.60·47-s + 1/7·49-s − 1.07·55-s − 1.58·57-s + 1.56·59-s − 0.768·61-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.11492829159469, −12.69385328910717, −12.17570388588064, −11.87681615459629, −11.37510689612406, −10.85261661130252, −10.16481973732528, −9.864575651534905, −9.211660955505425, −8.809181444040083, −8.315189810766669, −8.036469235518442, −7.650540221839253, −7.181550190648470, −6.766417943766744, −6.195291070242763, −5.076863652832844, −4.597636443533759, −4.319946239388332, −3.824710169847111, −3.271639432861018, −2.781261358038574, −2.324949789683820, −1.595887410202452, −0.8383405386554193, 0,
0.8383405386554193, 1.595887410202452, 2.324949789683820, 2.781261358038574, 3.271639432861018, 3.824710169847111, 4.319946239388332, 4.597636443533759, 5.076863652832844, 6.195291070242763, 6.766417943766744, 7.181550190648470, 7.650540221839253, 8.036469235518442, 8.315189810766669, 8.809181444040083, 9.211660955505425, 9.864575651534905, 10.16481973732528, 10.85261661130252, 11.37510689612406, 11.87681615459629, 12.17570388588064, 12.69385328910717, 13.11492829159469
