L(s) = 1 | − i·2-s + (1.41 + i)3-s − 4-s + (1.73 − 1.41i)5-s + (1 − 1.41i)6-s − 2.44i·7-s + i·8-s + (1.00 + 2.82i)9-s + (−1.41 − 1.73i)10-s − 1.41i·11-s + (−1.41 − i)12-s + 4.24·13-s − 2.44·14-s + (3.86 − 0.267i)15-s + 16-s − 6.92·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.816 + 0.577i)3-s − 0.5·4-s + (0.774 − 0.632i)5-s + (0.408 − 0.577i)6-s − 0.925i·7-s + 0.353i·8-s + (0.333 + 0.942i)9-s + (−0.447 − 0.547i)10-s − 0.426i·11-s + (−0.408 − 0.288i)12-s + 1.17·13-s − 0.654·14-s + (0.997 − 0.0691i)15-s + 0.250·16-s − 1.68·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.459 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.77046 - 1.07687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77046 - 1.07687i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| 5 | \( 1 + (-1.73 + 1.41i)T \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 - 6.92iT - 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 7.34iT - 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 4.89T + 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−10.65171534505724186868597983305, −9.702943753807862942401007006033, −8.896493302555917597315301269739, −8.440272275862917848297377120413, −7.11018685262908510218373045466, −5.76574549789113542450504647263, −4.58317267789085601599603807991, −3.87033497310690517431515260208, −2.62233891280913395759714076567, −1.28120733755846271928916825873,
1.82701818561583760460637274292, 2.90165344209995032257333272853, 4.24772602455232353809365655064, 5.85608243995193203622258917694, 6.29924481020775683412890395392, 7.28817821247907017148069853043, 8.212272003128964222119770684347, 9.107730096612012138912099287640, 9.559471824343587157143558040211, 10.79841660906424039730909438086