L(s) = 1 | + (1.5 + 2.59i)5-s + (−1 + 1.73i)7-s + (−3 + 5.19i)11-s + (−2.5 − 4.33i)13-s + 3·17-s + 2·19-s + (3 + 5.19i)23-s + (−2 + 3.46i)25-s + (1.5 − 2.59i)29-s + (2 + 3.46i)31-s − 6·35-s + 5·37-s + (−3 − 5.19i)41-s + (5 − 8.66i)43-s + (1.50 + 2.59i)49-s + ⋯ |
L(s) = 1 | + (0.670 + 1.16i)5-s + (−0.377 + 0.654i)7-s + (−0.904 + 1.56i)11-s + (−0.693 − 1.20i)13-s + 0.727·17-s + 0.458·19-s + (0.625 + 1.08i)23-s + (−0.400 + 0.692i)25-s + (0.278 − 0.482i)29-s + (0.359 + 0.622i)31-s − 1.01·35-s + 0.821·37-s + (−0.468 − 0.811i)41-s + (0.762 − 1.32i)43-s + (0.214 + 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.983282 + 0.825071i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.983282 + 0.825071i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.96297238117988667640298414736, −10.53374388284959934229764616553, −10.11664526213416518522555434750, −9.323598198158614980138775093847, −7.72041112268156637662184917747, −7.14427200571498360767566607545, −5.86595632836788444094026650577, −5.06591642150914633137430871306, −3.13604724997641892669080831300, −2.31321220030172474812161336887,
0.957563683851807472073539154453, 2.84814439494514772339290462718, 4.42296694578054834532579966763, 5.40080429617027325624458101214, 6.40285866834164375620609452158, 7.70411241119005063996128701604, 8.698233963198732900725570156511, 9.494133237303159615432348337833, 10.37864646771819563033577462290, 11.41470032453232554750095195641