| L(s) = 1 | − 3-s − 5-s − 3·7-s + 9-s + 3·11-s + 13-s + 15-s − 5·17-s + 3·21-s + 9·23-s + 25-s − 27-s − 29-s − 8·31-s − 3·33-s + 3·35-s − 37-s − 39-s − 5·41-s − 10·43-s − 45-s − 4·47-s + 2·49-s + 5·51-s − 9·53-s − 3·55-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.258·15-s − 1.21·17-s + 0.654·21-s + 1.87·23-s + 1/5·25-s − 0.192·27-s − 0.185·29-s − 1.43·31-s − 0.522·33-s + 0.507·35-s − 0.164·37-s − 0.160·39-s − 0.780·41-s − 1.52·43-s − 0.149·45-s − 0.583·47-s + 2/7·49-s + 0.700·51-s − 1.23·53-s − 0.404·55-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1689455639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1689455639\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
| 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
−12.59414189167864, −12.03740828466051, −11.51457888664996, −11.20555840984180, −10.88494661577070, −10.28369489277905, −9.782827164651211, −9.292082359282082, −8.972908900587151, −8.543806277579557, −7.950620183738345, −7.151506643329989, −6.836296965823692, −6.701511239208869, −6.117809530182369, −5.512927597035512, −4.990440220328434, −4.537152953281255, −3.921330413547107, −3.388538214830104, −3.163226991633130, −2.291823338283103, −1.594828623365383, −1.050640233494962, −0.1218754859315839,
0.1218754859315839, 1.050640233494962, 1.594828623365383, 2.291823338283103, 3.163226991633130, 3.388538214830104, 3.921330413547107, 4.537152953281255, 4.990440220328434, 5.512927597035512, 6.117809530182369, 6.701511239208869, 6.836296965823692, 7.151506643329989, 7.950620183738345, 8.543806277579557, 8.972908900587151, 9.292082359282082, 9.782827164651211, 10.28369489277905, 10.88494661577070, 11.20555840984180, 11.51457888664996, 12.03740828466051, 12.59414189167864
