L(s) = 1 | − i·2-s − 0.585i·3-s − 4-s − 0.585·6-s + i·8-s + 2.65·9-s + 4.82·11-s + 0.585i·12-s − 0.828i·13-s + 16-s + 5.41i·17-s − 2.65i·18-s − 3.41·19-s − 4.82i·22-s + 6.82i·23-s + 0.585·24-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.338i·3-s − 0.5·4-s − 0.239·6-s + 0.353i·8-s + 0.885·9-s + 1.45·11-s + 0.169i·12-s − 0.229i·13-s + 0.250·16-s + 1.31i·17-s − 0.626i·18-s − 0.783·19-s − 1.02i·22-s + 1.42i·23-s + 0.119·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.929117322\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.929117322\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 0.585iT - 3T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.828iT - 13T^{2} \) |
| 17 | \( 1 - 5.41iT - 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 - 6.82iT - 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 3.65iT - 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 3.17iT - 43T^{2} \) |
| 47 | \( 1 - 10.8iT - 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 9.65iT - 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 6.58iT - 73T^{2} \) |
| 79 | \( 1 - 1.17T + 79T^{2} \) |
| 83 | \( 1 + 6.24iT - 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 16.2iT - 97T^{2} \) |
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
−9.216775286300177306031598599612, −8.080914234065197501836451054057, −7.51843733102565442365396653653, −6.39399130804145172025830358645, −5.98611343077453752250912950188, −4.52926207254757590896453942987, −4.08159495995994266439823254636, −3.12584128703692414453406016990, −1.77503131842460740954774796660, −1.20516095868115670357061344905,
0.77166113878200625449024436728, 2.17954991242100634017009984744, 3.60473933700510773234109545827, 4.35300909641337723094536001973, 4.89893057642734328137393798047, 6.09143497799079737905451503493, 6.73825757635465499039326743097, 7.26084846068139564086523327622, 8.252169554957680857501380845436, 9.133188469852560898417838400406