L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.22 + 1.22i)3-s − 1.00i·4-s + (2.12 + 0.707i)5-s + 1.73i·6-s + (−3.44 − 3.44i)7-s + (−0.707 − 0.707i)8-s − 2.99i·9-s + (2 − 0.999i)10-s + 6.29i·11-s + (1.22 + 1.22i)12-s + (−4.44 + 4.44i)13-s − 4.87·14-s + (−3.46 + 1.73i)15-s − 1.00·16-s + (0.317 − 0.317i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.707 + 0.707i)3-s − 0.500i·4-s + (0.948 + 0.316i)5-s + 0.707i·6-s + (−1.30 − 1.30i)7-s + (−0.250 − 0.250i)8-s − 0.999i·9-s + (0.632 − 0.316i)10-s + 1.89i·11-s + (0.353 + 0.353i)12-s + (−1.23 + 1.23i)13-s − 1.30·14-s + (−0.894 + 0.447i)15-s − 0.250·16-s + (0.0770 − 0.0770i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.554861 + 0.701097i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.554861 + 0.701097i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 + (-2.12 - 0.707i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + (3.44 + 3.44i)T + 7iT^{2} \) |
| 11 | \( 1 - 6.29iT - 11T^{2} \) |
| 13 | \( 1 + (4.44 - 4.44i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.317 + 0.317i)T - 17iT^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 + (1.73 + 1.73i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 2.82iT - 41T^{2} \) |
| 43 | \( 1 + (2.44 - 2.44i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.65 - 5.65i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 0.449T + 61T^{2} \) |
| 67 | \( 1 + (3.55 + 3.55i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.19iT - 71T^{2} \) |
| 73 | \( 1 + (-4.89 + 4.89i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.898iT - 79T^{2} \) |
| 83 | \( 1 + (6.29 + 6.29i)T + 83iT^{2} \) |
| 89 | \( 1 + 4.38T + 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−10.15080306227817534491314029497, −9.720612193156635011589519933135, −9.482473668475102427024906164251, −7.27689499781925582382690995188, −6.73496882540020277903289819688, −6.01660585846980116257059441653, −4.68425208510219863439617324620, −4.29471755727131771840961597226, −3.07323154535132534428308313569, −1.71768726828355245297856330616,
0.36481427558342586239392273008, 2.46812306961489505570117488750, 3.13174949931045273321606582126, 5.19652934255900475422530723156, 5.53156521004457743036184035370, 6.20577426797978136951793860438, 6.87942089771188767096365770046, 8.151158633218816507483095922032, 8.850914917967499007857995654268, 9.772534613370680180533228469646