L(s) = 1 | + (1.73 + 0.0216i)3-s + (−2.64 + 0.0973i)7-s + (2.99 + 0.0749i)9-s + (2.62 + 1.51i)11-s − 2.31i·13-s + (−2.59 + 4.49i)17-s + (5.58 − 3.22i)19-s + (−4.58 + 0.111i)21-s + (4.21 − 2.43i)23-s + (5.19 + 0.194i)27-s + 3.48i·29-s + (1.16 + 0.673i)31-s + (4.52 + 2.68i)33-s + (−1.40 − 2.43i)37-s + (0.0501 − 4.01i)39-s + ⋯ |
L(s) = 1 | + (0.999 + 0.0124i)3-s + (−0.999 + 0.0368i)7-s + (0.999 + 0.0249i)9-s + (0.792 + 0.457i)11-s − 0.642i·13-s + (−0.629 + 1.09i)17-s + (1.28 − 0.739i)19-s + (−0.999 + 0.0243i)21-s + (0.879 − 0.507i)23-s + (0.999 + 0.0374i)27-s + 0.647i·29-s + (0.209 + 0.120i)31-s + (0.786 + 0.467i)33-s + (−0.231 − 0.400i)37-s + (0.00803 − 0.642i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.534727305\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.534727305\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.73 - 0.0216i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.64 - 0.0973i)T \) |
good | 11 | \( 1 + (-2.62 - 1.51i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.31iT - 13T^{2} \) |
| 17 | \( 1 + (2.59 - 4.49i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.58 + 3.22i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.21 + 2.43i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.48iT - 29T^{2} \) |
| 31 | \( 1 + (-1.16 - 0.673i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.40 + 2.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 1.06T + 41T^{2} \) |
| 43 | \( 1 - 2.42T + 43T^{2} \) |
| 47 | \( 1 + (-1.90 - 3.30i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.11 - 3.52i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.18 - 10.7i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-11.4 + 6.60i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.14 + 3.72i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.08iT - 71T^{2} \) |
| 73 | \( 1 + (-0.132 - 0.0763i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.04 - 5.27i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + (-7.10 - 12.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8.63iT - 97T^{2} \) |
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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
−9.043456466511106748967139071593, −8.598265024176261150525109778965, −7.46407445607579818887469022412, −6.95248414235225505836791345155, −6.15213734009649537833321020571, −4.98723660226435176820342747743, −3.99339982537238523758893856324, −3.26018957157399777302302266713, −2.43553900204711845193428725732, −1.10057245706385566677673145960,
1.01633919062938990123612420262, 2.35650959257809695266322488909, 3.30255786593561683399087942537, 3.87514797678927599412926306112, 4.96380011773719142120550544462, 6.09654416458702354533141068646, 6.95750046057606274360058952503, 7.38727821973517220033390544262, 8.510225522918374641370993747689, 9.148800814329543455763316552722