| L(s) = 1 | − 5·5-s − 2·7-s − 34·11-s − 68·13-s + 38·17-s − 4·19-s + 152·23-s + 25·25-s + 46·29-s + 260·31-s + 10·35-s − 312·37-s − 48·41-s + 200·43-s + 104·47-s − 339·49-s + 414·53-s + 170·55-s − 2·59-s − 38·61-s + 340·65-s + 244·67-s + 708·71-s − 378·73-s + 68·77-s + 852·79-s + 844·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.107·7-s − 0.931·11-s − 1.45·13-s + 0.542·17-s − 0.0482·19-s + 1.37·23-s + 1/5·25-s + 0.294·29-s + 1.50·31-s + 0.0482·35-s − 1.38·37-s − 0.182·41-s + 0.709·43-s + 0.322·47-s − 0.988·49-s + 1.07·53-s + 0.416·55-s − 0.00441·59-s − 0.0797·61-s + 0.648·65-s + 0.444·67-s + 1.18·71-s − 0.606·73-s + 0.100·77-s + 1.21·79-s + 1.11·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.378835983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.378835983\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 34 T + p^{3} T^{2} \) |
| 13 | \( 1 + 68 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 - 46 T + p^{3} T^{2} \) |
| 31 | \( 1 - 260 T + p^{3} T^{2} \) |
| 37 | \( 1 + 312 T + p^{3} T^{2} \) |
| 41 | \( 1 + 48 T + p^{3} T^{2} \) |
| 43 | \( 1 - 200 T + p^{3} T^{2} \) |
| 47 | \( 1 - 104 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 2 T + p^{3} T^{2} \) |
| 61 | \( 1 + 38 T + p^{3} T^{2} \) |
| 67 | \( 1 - 244 T + p^{3} T^{2} \) |
| 71 | \( 1 - 708 T + p^{3} T^{2} \) |
| 73 | \( 1 + 378 T + p^{3} T^{2} \) |
| 79 | \( 1 - 852 T + p^{3} T^{2} \) |
| 83 | \( 1 - 844 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1380 T + p^{3} T^{2} \) |
| 97 | \( 1 - 514 T + p^{3} T^{2} \) |
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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
−10.11187782250396200274996933909, −9.189121205191921281272103721032, −8.170146821112863584525485937680, −7.45764529737290360366412027284, −6.65000600802405336255556184818, −5.29201333312129347876388304177, −4.70975174807131255435401986548, −3.31070307408484721161973864201, −2.39165222229586030186304490517, −0.65328079464842706794257121661,
0.65328079464842706794257121661, 2.39165222229586030186304490517, 3.31070307408484721161973864201, 4.70975174807131255435401986548, 5.29201333312129347876388304177, 6.65000600802405336255556184818, 7.45764529737290360366412027284, 8.170146821112863584525485937680, 9.189121205191921281272103721032, 10.11187782250396200274996933909
