| L(s) = 1 | + 3-s − 1.49·5-s + 0.665i·7-s + 9-s − 0.592i·11-s + 1.37i·13-s − 1.49·15-s − 3.18·17-s + (−4.33 − 0.466i)19-s + 0.665i·21-s + 8.03i·23-s − 2.75·25-s + 27-s + 3.06i·29-s − 0.0945·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.669·5-s + 0.251i·7-s + 0.333·9-s − 0.178i·11-s + 0.381i·13-s − 0.386·15-s − 0.771·17-s + (−0.994 − 0.106i)19-s + 0.145i·21-s + 1.67i·23-s − 0.551·25-s + 0.192·27-s + 0.569i·29-s − 0.0169·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8897204681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8897204681\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + (4.33 + 0.466i)T \) |
| good | 5 | \( 1 + 1.49T + 5T^{2} \) |
| 7 | \( 1 - 0.665iT - 7T^{2} \) |
| 11 | \( 1 + 0.592iT - 11T^{2} \) |
| 13 | \( 1 - 1.37iT - 13T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 23 | \( 1 - 8.03iT - 23T^{2} \) |
| 29 | \( 1 - 3.06iT - 29T^{2} \) |
| 31 | \( 1 + 0.0945T + 31T^{2} \) |
| 37 | \( 1 - 3.62iT - 37T^{2} \) |
| 41 | \( 1 - 4.18iT - 41T^{2} \) |
| 43 | \( 1 + 10.7iT - 43T^{2} \) |
| 47 | \( 1 - 6.22iT - 47T^{2} \) |
| 53 | \( 1 - 4.18iT - 53T^{2} \) |
| 59 | \( 1 + 7.83T + 59T^{2} \) |
| 61 | \( 1 - 4.17T + 61T^{2} \) |
| 67 | \( 1 + 3.03T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 1.21T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 1.84iT - 83T^{2} \) |
| 89 | \( 1 - 6.73iT - 89T^{2} \) |
| 97 | \( 1 - 12.7iT - 97T^{2} \) |
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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
−9.275777635302338588324318610019, −8.822069033364483891555493982169, −7.985459902096376823652535503365, −7.31737976137325631166194567972, −6.48312743736265204922298027076, −5.48983384499087413119133008599, −4.39327271573890149993049546362, −3.74558418060682543263919356058, −2.70940846021588958186865937103, −1.60217916716156273642654240257,
0.29050836878567639106969184163, 1.99705889319239226703896665776, 2.94876548452491177689317934643, 4.14100014048823534390273787494, 4.46925258844864424323812683855, 5.85278900805170429187414475776, 6.74516214035685412720557001496, 7.48175733389485752538019734912, 8.316026646481719384680024530478, 8.731599243531765574725441671682
