L(s) = 1 | + (0.809 − 1.40i)2-s + (−1.11 + 1.93i)3-s + (−0.309 − 0.535i)4-s + (1.80 + 3.13i)6-s + (−0.118 − 0.204i)7-s + 2.23·8-s + (−1 − 1.73i)9-s + (−2.11 + 3.66i)11-s + 1.38·12-s + (1 + 3.46i)13-s − 0.381·14-s + (2.42 − 4.20i)16-s + (2.73 + 4.73i)17-s − 3.23·18-s + (0.118 + 0.204i)19-s + ⋯ |
L(s) = 1 | + (0.572 − 0.990i)2-s + (−0.645 + 1.11i)3-s + (−0.154 − 0.267i)4-s + (0.738 + 1.27i)6-s + (−0.0446 − 0.0772i)7-s + 0.790·8-s + (−0.333 − 0.577i)9-s + (−0.638 + 1.10i)11-s + 0.398·12-s + (0.277 + 0.960i)13-s − 0.102·14-s + (0.606 − 1.05i)16-s + (0.663 + 1.14i)17-s − 0.762·18-s + (0.0270 + 0.0469i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44859 + 0.398123i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44859 + 0.398123i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (-1 - 3.46i)T \) |
good | 2 | \( 1 + (-0.809 + 1.40i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.11 - 1.93i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (0.118 + 0.204i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.11 - 3.66i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.73 - 4.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.118 - 0.204i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.11 + 7.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.736 - 1.27i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.97 + 5.14i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.881 + 1.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (6.35 + 11.0i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.20 - 10.7i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.35 + 9.27i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.881 - 1.52i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.73 - 4.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.57763077785528128510305960317, −10.70499298177655336098048784137, −10.31674451286190025880640002540, −9.366455143692521586295709792065, −7.946667428119654024854326274085, −6.66629651106512238511901918883, −5.18610495781063016831296549462, −4.47698604952670919961962973509, −3.60236814137968309890456470238, −2.03214319651505842861828524135,
1.07479646069549256135632957733, 3.17444443030196359609939706780, 5.12048435712763670274353868291, 5.71608149209922709986782640188, 6.53467463696251984531624811190, 7.55347861868168264227253097549, 8.040107285992264161983795794839, 9.646197803898578719857196614600, 10.95412553674207765490254482214, 11.55828148208471593298850415137