L(s) = 1 | + 0.231·2-s + (−1.66 + 2.87i)3-s − 1.94·4-s + (−1.11 + 1.93i)5-s + (−0.384 + 0.665i)6-s − 0.913·8-s + (−4.01 − 6.96i)9-s + (−0.258 + 0.447i)10-s + (−1.66 + 2.87i)11-s + (3.23 − 5.60i)12-s + (−3.40 + 1.19i)13-s + (−3.70 − 6.41i)15-s + 3.68·16-s + 1.37·17-s + (−0.929 − 1.61i)18-s + (1.61 + 2.80i)19-s + ⋯ |
L(s) = 1 | + 0.163·2-s + (−0.959 + 1.66i)3-s − 0.973·4-s + (−0.498 + 0.864i)5-s + (−0.156 + 0.271i)6-s − 0.322·8-s + (−1.33 − 2.32i)9-s + (−0.0816 + 0.141i)10-s + (−0.500 + 0.867i)11-s + (0.933 − 1.61i)12-s + (−0.943 + 0.330i)13-s + (−0.957 − 1.65i)15-s + 0.920·16-s + 0.333·17-s + (−0.219 − 0.379i)18-s + (0.371 + 0.642i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.126109 - 0.0896410i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.126109 - 0.0896410i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (3.40 - 1.19i)T \) |
good | 2 | \( 1 - 0.231T + 2T^{2} \) |
| 3 | \( 1 + (1.66 - 2.87i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.11 - 1.93i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.66 - 2.87i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 + (-1.61 - 2.80i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 0.838T + 23T^{2} \) |
| 29 | \( 1 + (-0.303 - 0.525i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.857 - 1.48i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 + (4.58 + 7.94i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.615 - 1.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.814 - 1.41i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.19 + 7.27i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 + (-2.73 - 4.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.09 + 8.82i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.60 + 4.51i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.98 + 3.43i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.22 - 5.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4.64T + 83T^{2} \) |
| 89 | \( 1 + 9.12T + 89T^{2} \) |
| 97 | \( 1 + (7.67 - 13.3i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.07201147956892557852603260597, −10.26639436362180520545812472858, −9.816208600408797371273182378231, −9.072152276258394572586945792950, −7.79977863118772392303931209732, −6.64551248932714381739948705868, −5.41268484168231628624168043542, −4.86552681347448420102852176520, −3.97833439265206632768137301538, −3.14626964356967952620950900044,
0.11553328741105699222498847588, 1.04584439045967007931850604839, 2.87226972054584269675860941859, 4.68899178066703332277551039800, 5.28421602891361166572342331209, 6.11914656234842744851520882018, 7.33271205402128687909479868296, 8.068178119724280299131735513680, 8.627660532864009303907217465336, 9.903571289923274923687357877788