L(s) = 1 | + (−1.65 + 0.524i)3-s + (1.01 − 1.75i)5-s + (4.09 − 2.36i)7-s + (2.44 − 1.73i)9-s + (−2.93 + 1.69i)11-s + (5.51 + 3.18i)13-s + (−0.751 + 3.42i)15-s − 4.87i·17-s + 1.48·19-s + (−5.51 + 6.04i)21-s + (−0.751 + 1.30i)23-s + (0.449 + 0.778i)25-s + (−3.13 + 4.14i)27-s + (−3.49 − 6.04i)29-s + (−5.93 − 3.42i)31-s + ⋯ |
L(s) = 1 | + (−0.953 + 0.302i)3-s + (0.452 − 0.784i)5-s + (1.54 − 0.893i)7-s + (0.816 − 0.577i)9-s + (−0.883 + 0.510i)11-s + (1.53 + 0.883i)13-s + (−0.193 + 0.884i)15-s − 1.18i·17-s + 0.340·19-s + (−1.20 + 1.32i)21-s + (−0.156 + 0.271i)23-s + (0.0898 + 0.155i)25-s + (−0.603 + 0.797i)27-s + (−0.648 − 1.12i)29-s + (−1.06 − 0.615i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.573136408\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.573136408\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.65 - 0.524i)T \) |
good | 5 | \( 1 + (-1.01 + 1.75i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-4.09 + 2.36i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.93 - 1.69i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.51 - 3.18i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.87iT - 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 + (0.751 - 1.30i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.49 + 6.04i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.93 + 3.42i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.86iT - 37T^{2} \) |
| 41 | \( 1 + (-4.62 - 2.66i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.76 - 4.78i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.09 - 7.09i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 4.96T + 53T^{2} \) |
| 59 | \( 1 + (7.88 + 4.55i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.51 - 3.18i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.48 + 9.50i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.68T + 71T^{2} \) |
| 73 | \( 1 - 0.449T + 73T^{2} \) |
| 79 | \( 1 + (-5.93 + 3.42i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.06 + 3.50i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 0.142iT - 89T^{2} \) |
| 97 | \( 1 + (-0.724 - 1.25i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.615205817623803017454907373079, −9.126670106703510782032472957714, −7.84945781658673890466472965646, −7.36837286944502558714517398589, −6.11081670779403045354102673692, −5.28454085230537620770121256039, −4.64390606571161834559623696647, −3.94090952592795730889905703208, −1.86007286281715206881432669829, −0.922113514766020083477468826454,
1.33339783951642718632919514423, 2.36088166295061090778390965497, 3.74711066168139500478219946329, 5.22056013034253437047351017333, 5.59817191665945400193796060274, 6.30477825344495626019247289178, 7.45300311184326185091160352895, 8.228922716961726347018844283839, 8.850951734820128506580268867092, 10.49044936543857939989350461810