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 | \( 1 - 16.2iT - 97T^{2} \) |
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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
−9.133188469852560898417838400406, −8.252169554957680857501380845436, −7.26084846068139564086523327622, −6.73825757635465499039326743097, −6.09143497799079737905451503493, −4.89893057642734328137393798047, −4.35300909641337723094536001973, −3.60473933700510773234109545827, −2.17954991242100634017009984744, −0.77166113878200625449024436728,
1.20516095868115670357061344905, 1.77503131842460740954774796660, 3.12584128703692414453406016990, 4.08159495995994266439823254636, 4.52926207254757590896453942987, 5.98611343077453752250912950188, 6.39399130804145172025830358645, 7.51843733102565442365396653653, 8.080914234065197501836451054057, 9.216775286300177306031598599612