| 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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.023776018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.023776018\) |
| \(L(3)\) |
|
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 | \( 1 + 3383 T + p^{4} T^{2} \) |
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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
−10.90727281188623126209078062031, −10.02167324635198187947274643445, −9.452553622857044050523915139297, −8.117164459554557883416801561379, −6.77870098325526192159847564067, −6.35352979342973832824196640061, −5.27856527695171052019033031417, −3.47061394966558839380758885103, −2.45932937843407106308840415054, −1.13848213976879026999663938709,
1.13848213976879026999663938709, 2.45932937843407106308840415054, 3.47061394966558839380758885103, 5.27856527695171052019033031417, 6.35352979342973832824196640061, 6.77870098325526192159847564067, 8.117164459554557883416801561379, 9.452553622857044050523915139297, 10.02167324635198187947274643445, 10.90727281188623126209078062031
