| L(s) = 1 | + 14·5-s + 81·9-s + 238·13-s + 322·17-s − 429·25-s − 82·29-s − 2.16e3·37-s − 3.03e3·41-s + 1.13e3·45-s + 2.40e3·49-s − 2.48e3·53-s + 6.95e3·61-s + 3.33e3·65-s + 1.44e3·73-s + 6.56e3·81-s + 4.50e3·85-s − 9.75e3·89-s − 1.91e3·97-s − 1.88e4·101-s − 9.36e3·109-s − 2.46e4·113-s + 1.92e4·117-s + ⋯ |
| L(s) = 1 | + 0.559·5-s + 9-s + 1.40·13-s + 1.11·17-s − 0.686·25-s − 0.0975·29-s − 1.57·37-s − 1.80·41-s + 0.559·45-s + 49-s − 0.883·53-s + 1.86·61-s + 0.788·65-s + 0.270·73-s + 81-s + 0.623·85-s − 1.23·89-s − 0.203·97-s − 1.84·101-s − 0.787·109-s − 1.92·113-s + 1.40·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.902585161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902585161\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 5 | \( 1 - 14 T + p^{4} T^{2} \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 - 238 T + p^{4} T^{2} \) |
| 17 | \( 1 - 322 T + p^{4} T^{2} \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( 1 + 82 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( 1 + 2162 T + p^{4} T^{2} \) |
| 41 | \( 1 + 3038 T + p^{4} T^{2} \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( 1 + 2482 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 - 6958 T + p^{4} T^{2} \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 - 1442 T + p^{4} T^{2} \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( 1 + 9758 T + p^{4} T^{2} \) |
| 97 | \( 1 + 1918 T + p^{4} T^{2} \) |
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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
−13.98854262116414467487283372785, −13.19948637567232317859455332659, −12.01976340164903850504526174401, −10.60458130807511105456361997135, −9.681891795913749497326036331059, −8.301220511703890261401576500805, −6.83525678987165971068821721965, −5.49397901031855704077965920800, −3.71181906572610660469945677070, −1.49998087539089393848938004561,
1.49998087539089393848938004561, 3.71181906572610660469945677070, 5.49397901031855704077965920800, 6.83525678987165971068821721965, 8.301220511703890261401576500805, 9.681891795913749497326036331059, 10.60458130807511105456361997135, 12.01976340164903850504526174401, 13.19948637567232317859455332659, 13.98854262116414467487283372785
