| 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 | − 1.88·3-s + 4-s + 5-s + 7-s + 2.56·9-s − 0.603·11-s − 1.88·12-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 − 1.88·48-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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.266183003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.266183003\) |
| \(L(3)\) |
|
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 | \( 1 - 3383 T + p^{4} T^{2} \) |
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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
−16.19474744889754003590654566389, −14.83482097907433557784164830239, −13.00461444216975226578297848492, −11.83069171515631438639831883790, −10.94512562091500552693882034186, −10.10450979160767154566662577625, −7.43528368594221714724175268990, −6.05265221425787203097340966688, −5.18837178119477882207412223984, −1.53470705928359760554534497106,
1.53470705928359760554534497106, 5.18837178119477882207412223984, 6.05265221425787203097340966688, 7.43528368594221714724175268990, 10.10450979160767154566662577625, 10.94512562091500552693882034186, 11.83069171515631438639831883790, 13.00461444216975226578297848492, 14.83482097907433557784164830239, 16.19474744889754003590654566389
