| L(s) = 1 | + 17·3-s + 16·4-s − 25·5-s − 49·7-s + 208·9-s − 73·11-s + 272·12-s − 23·13-s − 425·15-s + 256·16-s − 263·17-s − 400·20-s − 833·21-s + 625·25-s + 2.15e3·27-s − 784·28-s − 1.15e3·29-s − 1.24e3·33-s + 1.22e3·35-s + 3.32e3·36-s − 391·39-s − 1.16e3·44-s − 5.20e3·45-s + 3.45e3·47-s + 4.35e3·48-s + 2.40e3·49-s − 4.47e3·51-s + ⋯ |
| L(s) = 1 | + 17/9·3-s + 4-s − 5-s − 7-s + 2.56·9-s − 0.603·11-s + 17/9·12-s − 0.136·13-s − 1.88·15-s + 16-s − 0.910·17-s − 20-s − 1.88·21-s + 25-s + 2.96·27-s − 28-s − 1.37·29-s − 1.13·33-s + 35-s + 2.56·36-s − 0.257·39-s − 0.603·44-s − 2.56·45-s + 1.56·47-s + 17/9·48-s + 49-s − 1.71·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.328569801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.328569801\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 2 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 3 | \( 1 - 17 T + p^{4} T^{2} \) |
| 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
−15.51487124230517903403246716626, −14.94843346995382516901317179392, −13.41378277649849048883335845106, −12.40308441785771029161524693376, −10.65105576750368463917003000635, −9.213682664261132612134472079807, −7.889029152079680423832801040629, −6.97696556564300181827071952644, −3.73081912494297591235812834728, −2.53257142807090831180213907965,
2.53257142807090831180213907965, 3.73081912494297591235812834728, 6.97696556564300181827071952644, 7.889029152079680423832801040629, 9.213682664261132612134472079807, 10.65105576750368463917003000635, 12.40308441785771029161524693376, 13.41378277649849048883335845106, 14.94843346995382516901317179392, 15.51487124230517903403246716626
