L(s) = 1 | + (−0.5 − 0.866i)2-s + (1 − 1.73i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 1.99·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (2 − 3.46i)11-s + (0.999 + 1.73i)12-s − 2·13-s + 1.99·15-s + (−0.5 − 0.866i)16-s + (4 − 6.92i)17-s + (−0.499 + 0.866i)18-s + (−3 − 5.19i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.816·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.603 − 1.04i)11-s + (0.288 + 0.499i)12-s − 0.554·13-s + 0.516·15-s + (−0.125 − 0.216i)16-s + (0.970 − 1.68i)17-s + (−0.117 + 0.204i)18-s + (−0.688 − 1.19i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.802869 - 1.20699i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.802869 - 1.20699i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-4 + 6.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5 - 8.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7 - 12.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.84560147229775295656698611187, −9.572787442178669955368542586524, −9.059719114781702912993424005405, −7.87375476895947367204363792169, −7.30782213420336423466350458498, −6.28748599496345805338779337871, −4.86540560096281629856743900210, −3.22382974136967459542096115376, −2.47948227997659415759311738769, −1.01742070539340444173550624631,
1.81053271716598292592258808492, 3.71925177400430442454550325902, 4.46190645142353511998086055316, 5.61181738674530446664715174184, 6.66507668183281802319014349940, 7.85786156934055558862000400924, 8.627122960512358616067082236441, 9.511833543183468081187391431418, 10.01959710149517671502315903855, 10.76752401879459906178159978455