| L(s) = 1 | − 94·5-s + 144·7-s + 380·11-s + 814·13-s + 862·17-s − 1.15e3·19-s + 488·23-s + 5.71e3·25-s + 5.46e3·29-s + 9.56e3·31-s − 1.35e4·35-s − 1.05e4·37-s + 5.19e3·41-s − 1.70e4·43-s − 3.16e3·47-s + 3.92e3·49-s + 2.47e4·53-s − 3.57e4·55-s − 1.73e4·59-s + 4.36e3·61-s − 7.65e4·65-s − 5.28e3·67-s − 8.36e3·71-s + 3.94e4·73-s + 5.47e4·77-s + 4.23e4·79-s + 6.18e4·83-s + ⋯ |
| L(s) = 1 | − 1.68·5-s + 1.11·7-s + 0.946·11-s + 1.33·13-s + 0.723·17-s − 0.734·19-s + 0.192·23-s + 1.82·25-s + 1.20·29-s + 1.78·31-s − 1.86·35-s − 1.26·37-s + 0.482·41-s − 1.40·43-s − 0.209·47-s + 0.233·49-s + 1.21·53-s − 1.59·55-s − 0.650·59-s + 0.150·61-s − 2.24·65-s − 0.143·67-s − 0.196·71-s + 0.866·73-s + 1.05·77-s + 0.763·79-s + 0.985·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.595085700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.595085700\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 94 T + p^{5} T^{2} \) |
| 7 | \( 1 - 144 T + p^{5} T^{2} \) |
| 11 | \( 1 - 380 T + p^{5} T^{2} \) |
| 13 | \( 1 - 814 T + p^{5} T^{2} \) |
| 17 | \( 1 - 862 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1156 T + p^{5} T^{2} \) |
| 23 | \( 1 - 488 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5466 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9560 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10506 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5190 T + p^{5} T^{2} \) |
| 43 | \( 1 + 17084 T + p^{5} T^{2} \) |
| 47 | \( 1 + 3168 T + p^{5} T^{2} \) |
| 53 | \( 1 - 24770 T + p^{5} T^{2} \) |
| 59 | \( 1 + 17380 T + p^{5} T^{2} \) |
| 61 | \( 1 - 4366 T + p^{5} T^{2} \) |
| 67 | \( 1 + 5284 T + p^{5} T^{2} \) |
| 71 | \( 1 + 8360 T + p^{5} T^{2} \) |
| 73 | \( 1 - 39466 T + p^{5} T^{2} \) |
| 79 | \( 1 - 42376 T + p^{5} T^{2} \) |
| 83 | \( 1 - 61828 T + p^{5} T^{2} \) |
| 89 | \( 1 - 63078 T + p^{5} T^{2} \) |
| 97 | \( 1 + 16318 T + p^{5} T^{2} \) |
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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
−13.81339899569042225052700670260, −12.14784214769981728375455910323, −11.59261789426695131522342490401, −10.60886607044387779590233300346, −8.599686339082464029602411009338, −8.057833497902982451039273937820, −6.62129353366093994782944609610, −4.66143598610920607019803022522, −3.59308361067321220463224651142, −1.07304450437721904211001094366,
1.07304450437721904211001094366, 3.59308361067321220463224651142, 4.66143598610920607019803022522, 6.62129353366093994782944609610, 8.057833497902982451039273937820, 8.599686339082464029602411009338, 10.60886607044387779590233300346, 11.59261789426695131522342490401, 12.14784214769981728375455910323, 13.81339899569042225052700670260
