L(s) = 1 | + 3-s + (1 − 1.73i)4-s + (2.5 − 0.866i)7-s + 9-s − 6·11-s + (1 − 1.73i)12-s + (1 + 3.46i)13-s + (−1.99 − 3.46i)16-s + (3 − 5.19i)17-s − 19-s + (2.5 − 0.866i)21-s + (3 + 5.19i)23-s + (2.5 + 4.33i)25-s + 27-s + (1.00 − 5.19i)28-s + (−3 + 5.19i)29-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (0.5 − 0.866i)4-s + (0.944 − 0.327i)7-s + 0.333·9-s − 1.80·11-s + (0.288 − 0.499i)12-s + (0.277 + 0.960i)13-s + (−0.499 − 0.866i)16-s + (0.727 − 1.26i)17-s − 0.229·19-s + (0.545 − 0.188i)21-s + (0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s + 0.192·27-s + (0.188 − 0.981i)28-s + (−0.557 + 0.964i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.62453 - 0.536125i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62453 - 0.536125i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
| 13 | \( 1 + (-1 - 3.46i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 71 | \( 1 + (-6 - 10.3i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.41713663638587335348598029106, −10.99182368036093911821271645413, −9.925866125853771218934926643199, −9.055100404151577712676096623760, −7.69134143290365378570083879270, −7.18651121217718707186110819940, −5.53585830801100031380296469818, −4.79153412721784858462342708317, −2.94397027629789813849841049066, −1.58228865631821683910614440088,
2.21528582487949343099019962483, 3.23221464868058999456757382845, 4.71183318939704073392095496173, 5.98026603602669110992264905323, 7.55690675277842499043507175633, 8.078030100597531039402791895284, 8.694531887478948531156843407273, 10.51810372628278857101993379866, 10.76336730097982592066630913679, 12.29091137516407948144273983484