| L(s) = 1 | + 3-s − 4·5-s + 7-s + 9-s − 2·11-s − 2·13-s − 4·15-s − 4·19-s + 21-s − 6·23-s + 11·25-s + 27-s − 10·29-s − 8·31-s − 2·33-s − 4·35-s + 10·37-s − 2·39-s − 4·41-s − 8·43-s − 4·45-s − 4·47-s + 49-s + 10·53-s + 8·55-s − 4·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 1.03·15-s − 0.917·19-s + 0.218·21-s − 1.25·23-s + 11/5·25-s + 0.192·27-s − 1.85·29-s − 1.43·31-s − 0.348·33-s − 0.676·35-s + 1.64·37-s − 0.320·39-s − 0.624·41-s − 1.21·43-s − 0.596·45-s − 0.583·47-s + 1/7·49-s + 1.37·53-s + 1.07·55-s − 0.529·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−10.11222933407131620152148251461, −9.016474375521408712679261706335, −8.050306287517390944376390946650, −7.77278723062985727351711147882, −6.85036368281080789512161503853, −5.31505211056085532901675130626, −4.21546713870310250114993857360, −3.58625452137859596295479508108, −2.19264027338821122800426191253, 0,
2.19264027338821122800426191253, 3.58625452137859596295479508108, 4.21546713870310250114993857360, 5.31505211056085532901675130626, 6.85036368281080789512161503853, 7.77278723062985727351711147882, 8.050306287517390944376390946650, 9.016474375521408712679261706335, 10.11222933407131620152148251461
