| L(s) = 1 | − 49·7-s + 81·9-s + 206·11-s + 734·23-s + 625·25-s + 1.23e3·29-s − 1.29e3·37-s + 334·43-s + 2.40e3·49-s − 5.58e3·53-s − 3.96e3·63-s − 4.94e3·67-s − 2.91e3·71-s − 1.00e4·77-s + 3.64e3·79-s + 6.56e3·81-s + 1.66e4·99-s − 1.16e4·107-s − 1.25e4·109-s + 2.37e4·113-s + ⋯ |
| L(s) = 1 | − 7-s + 9-s + 1.70·11-s + 1.38·23-s + 25-s + 1.46·29-s − 0.945·37-s + 0.180·43-s + 49-s − 1.98·53-s − 63-s − 1.10·67-s − 0.578·71-s − 1.70·77-s + 0.584·79-s + 81-s + 1.70·99-s − 1.02·107-s − 1.05·109-s + 1.85·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.848523107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.848523107\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( 1 - 206 T + p^{4} T^{2} \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( 1 - 734 T + p^{4} T^{2} \) |
| 29 | \( 1 - 1234 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( 1 + 1294 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 - 334 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( 1 + 5582 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 + 4946 T + p^{4} T^{2} \) |
| 71 | \( 1 + 2914 T + p^{4} T^{2} \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( 1 - 3646 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
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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
−12.79997163617265986017428728488, −12.07800418326718200635217385918, −10.72187072399263487989917910217, −9.630897262914318321907109801694, −8.832230429710822922826940781329, −7.05838280237570572743596438549, −6.42523227218967556675364158034, −4.58603717500816176101938025165, −3.27609022170017089909666574010, −1.16647602157514645741976515280,
1.16647602157514645741976515280, 3.27609022170017089909666574010, 4.58603717500816176101938025165, 6.42523227218967556675364158034, 7.05838280237570572743596438549, 8.832230429710822922826940781329, 9.630897262914318321907109801694, 10.72187072399263487989917910217, 12.07800418326718200635217385918, 12.79997163617265986017428728488
