| L(s) = 1 | − i·3-s − 2.12·5-s + (2.43 + 1.03i)7-s − 9-s − 1.02i·11-s + 1.76i·13-s + 2.12i·15-s + 0.210·17-s − 4.64·19-s + (1.03 − 2.43i)21-s + (3.56 + 3.20i)23-s − 0.474·25-s + i·27-s + 10.2·29-s − 1.19i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.951·5-s + (0.919 + 0.392i)7-s − 0.333·9-s − 0.308i·11-s + 0.488i·13-s + 0.549i·15-s + 0.0510·17-s − 1.06·19-s + (0.226 − 0.531i)21-s + (0.742 + 0.669i)23-s − 0.0948·25-s + 0.192i·27-s + 1.89·29-s − 0.213i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.409042227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.409042227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-2.43 - 1.03i)T \) |
| 23 | \( 1 + (-3.56 - 3.20i)T \) |
| good | 5 | \( 1 + 2.12T + 5T^{2} \) |
| 11 | \( 1 + 1.02iT - 11T^{2} \) |
| 13 | \( 1 - 1.76iT - 13T^{2} \) |
| 17 | \( 1 - 0.210T + 17T^{2} \) |
| 19 | \( 1 + 4.64T + 19T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 1.19iT - 31T^{2} \) |
| 37 | \( 1 - 10.7iT - 37T^{2} \) |
| 41 | \( 1 + 1.14iT - 41T^{2} \) |
| 43 | \( 1 - 9.64iT - 43T^{2} \) |
| 47 | \( 1 + 5.56iT - 47T^{2} \) |
| 53 | \( 1 + 1.32iT - 53T^{2} \) |
| 59 | \( 1 + 9.90iT - 59T^{2} \) |
| 61 | \( 1 - 2.05T + 61T^{2} \) |
| 67 | \( 1 - 2.73iT - 67T^{2} \) |
| 71 | \( 1 - 6.57T + 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 1.45iT - 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 - 6.85T + 89T^{2} \) |
| 97 | \( 1 - 1.67T + 97T^{2} \) |
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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
−8.954175130786811998321598541416, −8.239877299837816148550327475505, −7.941936263679186029265493259480, −6.88817982642598412238134331886, −6.26002427383381126727865999218, −5.08357774724122847183759540494, −4.42975440159296286401568771130, −3.33656177128512147376316301502, −2.24408392696082213244433431086, −1.05764238671301702135978971533,
0.62816330850345866537545317187, 2.24572468791353152414503568144, 3.43611318646533844067315811186, 4.36020096659458566023196259068, 4.75614438372873057870947807755, 5.85020806277135041010915602201, 6.95061363505327670004845319357, 7.67184606096035110019003969886, 8.414630098669653746594838728634, 8.913904470166948250089397951128
