L(s) = 1 | + (0.5 + 0.866i)2-s + (−1 − 1.41i)3-s + (−0.499 + 0.866i)4-s + (0.724 − 1.57i)6-s + (0.224 + 0.389i)7-s − 0.999·8-s + (−1.00 + 2.82i)9-s + (2.44 + 4.24i)11-s + (1.72 − 0.158i)12-s + (0.224 − 0.389i)13-s + (−0.224 + 0.389i)14-s + (−0.5 − 0.866i)16-s + 4.89·17-s + (−2.94 + 0.548i)18-s + 7.44·19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.577 − 0.816i)3-s + (−0.249 + 0.433i)4-s + (0.295 − 0.642i)6-s + (0.0849 + 0.147i)7-s − 0.353·8-s + (−0.333 + 0.942i)9-s + (0.738 + 1.27i)11-s + (0.497 − 0.0458i)12-s + (0.0623 − 0.107i)13-s + (−0.0600 + 0.104i)14-s + (−0.125 − 0.216i)16-s + 1.18·17-s + (−0.695 + 0.129i)18-s + 1.70·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27384 + 0.601177i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27384 + 0.601177i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.224 - 0.389i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.44 - 4.24i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.224 + 0.389i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.22 - 3.85i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.27 + 2.20i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.44 - 9.43i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 + (-2.72 + 4.71i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.174 - 0.301i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + (8.34 + 14.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.72 - 4.71i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + (4.39 + 7.61i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.59384541821252761302908530306, −10.32641194812099078548895214717, −9.346874600038060300780673848000, −8.160501770309707555186817493691, −7.27639666692337619615276747302, −6.72059832324328362776879037984, −5.51939764232879064184959526712, −4.84679228538997208371004382170, −3.26185493975143258739506416654, −1.49089697440882793314451398627,
1.01458189923360101643839017856, 3.25437106887929030614686478107, 3.85628211549987466967111300548, 5.30341507107101600564116017366, 5.77436337087557809150953141771, 7.11553883721982001271986254863, 8.574144850049826890897421235758, 9.437185037028948740808429909465, 10.19518229786274596836568361561, 11.07196187741515230192585947817