L(s) = 1 | + (−1.00 + 1.41i)3-s + (1.92 − 1.11i)5-s + (1.98 + 1.74i)7-s + (−0.981 − 2.83i)9-s + (3.11 + 1.79i)11-s + (−0.699 − 0.404i)13-s + (−0.365 + 3.82i)15-s + (1.37 − 0.796i)17-s + (1.63 − 2.83i)19-s + (−4.46 + 1.05i)21-s + (1.46 − 0.844i)23-s + (−0.0343 + 0.0595i)25-s + (4.98 + 1.46i)27-s + (−2.13 − 3.69i)29-s + 1.26·31-s + ⋯ |
L(s) = 1 | + (−0.579 + 0.814i)3-s + (0.860 − 0.496i)5-s + (0.751 + 0.659i)7-s + (−0.327 − 0.944i)9-s + (0.939 + 0.542i)11-s + (−0.194 − 0.112i)13-s + (−0.0942 + 0.988i)15-s + (0.334 − 0.193i)17-s + (0.375 − 0.649i)19-s + (−0.973 + 0.229i)21-s + (0.305 − 0.176i)23-s + (−0.00687 + 0.0119i)25-s + (0.959 + 0.281i)27-s + (−0.395 − 0.685i)29-s + 0.226·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.761327138\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.761327138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.00 - 1.41i)T \) |
| 7 | \( 1 + (-1.98 - 1.74i)T \) |
good | 5 | \( 1 + (-1.92 + 1.11i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.11 - 1.79i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.699 + 0.404i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.37 + 0.796i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 2.83i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.46 + 0.844i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.13 + 3.69i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 + (-3.94 + 6.83i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.57 - 4.37i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (7.47 - 4.31i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 0.665T + 47T^{2} \) |
| 53 | \( 1 + (-6.27 - 10.8i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 4.30T + 59T^{2} \) |
| 61 | \( 1 + 8.59iT - 61T^{2} \) |
| 67 | \( 1 - 0.102iT - 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 + (5.30 - 3.06i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 4.31iT - 79T^{2} \) |
| 83 | \( 1 + (-5.33 - 9.23i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.69 - 0.981i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.3 + 8.84i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.772610479941193043697093799070, −9.426500409629572379849729739190, −8.746960236423055372947461548258, −7.55656461118735327805373845073, −6.36912619266270165345018844949, −5.61444637677482607118944957851, −4.92083023909959395488929553519, −4.11459817238410290061482525488, −2.60921554355727616294124962784, −1.24462808144468192119226480131,
1.11942063539002821846196458220, 2.04402473279739957732875137089, 3.48966606498906026558775691241, 4.81235903478217552366676325972, 5.76766477418993049954439852534, 6.44999392971863024165497004021, 7.24576982143082518207115255368, 8.026618847345753685526671851874, 9.003438351229241467423734997077, 10.13615516637901075570095855181