| L(s) = 1 | − 4·2-s + 16·4-s − 64·8-s + 46·11-s + 256·16-s + 574·17-s + 434·19-s − 184·22-s + 625·25-s − 1.02e3·32-s − 2.29e3·34-s − 1.73e3·38-s + 1.24e3·41-s − 3.50e3·43-s + 736·44-s + 2.40e3·49-s − 2.50e3·50-s + 238·59-s + 4.09e3·64-s − 5.13e3·67-s + 9.18e3·68-s + 9.50e3·73-s + 6.94e3·76-s − 4.98e3·82-s − 1.11e4·83-s + 1.40e4·86-s − 2.94e3·88-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s + 0.380·11-s + 16-s + 1.98·17-s + 1.20·19-s − 0.380·22-s + 25-s − 32-s − 1.98·34-s − 1.20·38-s + 0.741·41-s − 1.89·43-s + 0.380·44-s + 49-s − 50-s + 0.0683·59-s + 64-s − 1.14·67-s + 1.98·68-s + 1.78·73-s + 1.20·76-s − 0.741·82-s − 1.62·83-s + 1.89·86-s − 0.380·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.121519889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.121519889\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( 1 - 46 T + p^{4} T^{2} \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( 1 - 574 T + p^{4} T^{2} \) |
| 19 | \( 1 - 434 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( 1 - 1246 T + p^{4} T^{2} \) |
| 43 | \( 1 + 3502 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( 1 - 238 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 + 5134 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 - 9506 T + p^{4} T^{2} \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( 1 + 11186 T + p^{4} T^{2} \) |
| 89 | \( 1 + 5474 T + p^{4} T^{2} \) |
| 97 | \( 1 + 9982 T + p^{4} T^{2} \) |
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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
−14.10814446486934802731230603310, −12.43744612041662292594077455803, −11.58199475585457548701835733177, −10.29712167486445104576401407754, −9.403598517774178191079347476242, −8.121119847953233467091553569821, −7.04691269205404191139133604801, −5.56228531970033939795410780887, −3.19349166742238189649177998603, −1.14250040043129782542659165686,
1.14250040043129782542659165686, 3.19349166742238189649177998603, 5.56228531970033939795410780887, 7.04691269205404191139133604801, 8.121119847953233467091553569821, 9.403598517774178191079347476242, 10.29712167486445104576401407754, 11.58199475585457548701835733177, 12.43744612041662292594077455803, 14.10814446486934802731230603310
