| L(s) = 1 | + 16·4-s − 25·5-s + 49·7-s + 73·11-s + 23·13-s + 256·16-s − 263·17-s − 400·20-s + 625·25-s + 784·28-s + 1.15e3·29-s − 1.22e3·35-s + 1.16e3·44-s + 3.45e3·47-s + 2.40e3·49-s + 368·52-s − 1.82e3·55-s + 4.09e3·64-s − 575·65-s − 4.20e3·68-s + 1.00e4·71-s − 9.50e3·73-s + 3.57e3·77-s + 1.21e4·79-s − 6.40e3·80-s + 6.38e3·83-s + 6.57e3·85-s + ⋯ |
| L(s) = 1 | + 4-s − 5-s + 7-s + 0.603·11-s + 0.136·13-s + 16-s − 0.910·17-s − 20-s + 25-s + 28-s + 1.37·29-s − 35-s + 0.603·44-s + 1.56·47-s + 49-s + 0.136·52-s − 0.603·55-s + 64-s − 0.136·65-s − 0.910·68-s + 1.99·71-s − 1.78·73-s + 0.603·77-s + 1.94·79-s − 80-s + 0.926·83-s + 0.910·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.529293729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.529293729\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 2 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( 1 - 73 T + p^{4} T^{2} \) |
| 13 | \( 1 - 23 T + p^{4} T^{2} \) |
| 17 | \( 1 + 263 T + p^{4} T^{2} \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( 1 - 1153 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( 1 - 3457 T + p^{4} T^{2} \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( 1 - 10078 T + p^{4} T^{2} \) |
| 73 | \( 1 + 9502 T + p^{4} T^{2} \) |
| 79 | \( 1 - 12167 T + p^{4} T^{2} \) |
| 83 | \( 1 - 6382 T + p^{4} T^{2} \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 - 3383 T + p^{4} T^{2} \) |
| 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
−11.14169827501417819453778426814, −10.46862084264821887576153723480, −8.904788044812155669810465425491, −8.063583656677963155937712525887, −7.21254494712299882301260948405, −6.32109073254335625819948973452, −4.87494519743366996133717786272, −3.80023292654454020295098033991, −2.39948374559678777967862535598, −1.02430201843410146653340870684,
1.02430201843410146653340870684, 2.39948374559678777967862535598, 3.80023292654454020295098033991, 4.87494519743366996133717786272, 6.32109073254335625819948973452, 7.21254494712299882301260948405, 8.063583656677963155937712525887, 8.904788044812155669810465425491, 10.46862084264821887576153723480, 11.14169827501417819453778426814
