| L(s) = 1 | + 9·3-s − 34·5-s − 49·7-s + 81·9-s + 756·11-s + 678·13-s − 306·15-s − 1.83e3·17-s − 604·19-s − 441·21-s − 2.84e3·23-s − 1.96e3·25-s + 729·27-s + 6.87e3·29-s − 3.56e3·31-s + 6.80e3·33-s + 1.66e3·35-s + 1.45e4·37-s + 6.10e3·39-s + 5.96e3·41-s + 676·43-s − 2.75e3·45-s + 2.08e4·47-s + 2.40e3·49-s − 1.65e4·51-s + 3.23e4·53-s − 2.57e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.608·5-s − 0.377·7-s + 1/3·9-s + 1.88·11-s + 1.11·13-s − 0.351·15-s − 1.54·17-s − 0.383·19-s − 0.218·21-s − 1.11·23-s − 0.630·25-s + 0.192·27-s + 1.51·29-s − 0.666·31-s + 1.08·33-s + 0.229·35-s + 1.75·37-s + 0.642·39-s + 0.553·41-s + 0.0557·43-s − 0.202·45-s + 1.37·47-s + 1/7·49-s − 0.890·51-s + 1.58·53-s − 1.14·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.447499237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.447499237\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 34 T + p^{5} T^{2} \) |
| 11 | \( 1 - 756 T + p^{5} T^{2} \) |
| 13 | \( 1 - 678 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1838 T + p^{5} T^{2} \) |
| 19 | \( 1 + 604 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2840 T + p^{5} T^{2} \) |
| 29 | \( 1 - 6878 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3568 T + p^{5} T^{2} \) |
| 37 | \( 1 - 14598 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5962 T + p^{5} T^{2} \) |
| 43 | \( 1 - 676 T + p^{5} T^{2} \) |
| 47 | \( 1 - 20800 T + p^{5} T^{2} \) |
| 53 | \( 1 - 32390 T + p^{5} T^{2} \) |
| 59 | \( 1 + 42948 T + p^{5} T^{2} \) |
| 61 | \( 1 - 44806 T + p^{5} T^{2} \) |
| 67 | \( 1 - 39708 T + p^{5} T^{2} \) |
| 71 | \( 1 - 25800 T + p^{5} T^{2} \) |
| 73 | \( 1 - 58954 T + p^{5} T^{2} \) |
| 79 | \( 1 - 77648 T + p^{5} T^{2} \) |
| 83 | \( 1 + 35964 T + p^{5} T^{2} \) |
| 89 | \( 1 - 80842 T + p^{5} T^{2} \) |
| 97 | \( 1 + 64334 T + p^{5} T^{2} \) |
| 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
−10.83420725395007403939577049758, −9.544592351869769992861253074762, −8.858734297539502831434777644829, −8.067410302549124809829355982349, −6.77085844801391678473910000161, −6.16600220458897682842567514033, −4.14360248899998408942243314911, −3.87702871492437634699607488040, −2.26220533952713331031536856392, −0.859544380382042987934419832530,
0.859544380382042987934419832530, 2.26220533952713331031536856392, 3.87702871492437634699607488040, 4.14360248899998408942243314911, 6.16600220458897682842567514033, 6.77085844801391678473910000161, 8.067410302549124809829355982349, 8.858734297539502831434777644829, 9.544592351869769992861253074762, 10.83420725395007403939577049758
