L(s) = 1 | + 1.44i·2-s + 0.0382·3-s − 0.100·4-s − 3.85i·5-s + 0.0554i·6-s − 1.66i·7-s + 2.75i·8-s − 2.99·9-s + 5.59·10-s − 1.88i·11-s − 0.00384·12-s + (1.08 − 3.43i)13-s + 2.40·14-s − 0.147i·15-s − 4.19·16-s + 3.75·17-s + ⋯ |
L(s) = 1 | + 1.02i·2-s + 0.0220·3-s − 0.0503·4-s − 1.72i·5-s + 0.0226i·6-s − 0.628i·7-s + 0.973i·8-s − 0.999·9-s + 1.76·10-s − 0.566i·11-s − 0.00111·12-s + (0.301 − 0.953i)13-s + 0.643·14-s − 0.0381i·15-s − 1.04·16-s + 0.909·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37363 - 0.211729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37363 - 0.211729i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (-1.08 + 3.43i)T \) |
| 31 | \( 1 + iT \) |
good | 2 | \( 1 - 1.44iT - 2T^{2} \) |
| 3 | \( 1 - 0.0382T + 3T^{2} \) |
| 5 | \( 1 + 3.85iT - 5T^{2} \) |
| 7 | \( 1 + 1.66iT - 7T^{2} \) |
| 11 | \( 1 + 1.88iT - 11T^{2} \) |
| 17 | \( 1 - 3.75T + 17T^{2} \) |
| 19 | \( 1 + 3.26iT - 19T^{2} \) |
| 23 | \( 1 - 5.31T + 23T^{2} \) |
| 29 | \( 1 - 3.11T + 29T^{2} \) |
| 37 | \( 1 - 11.5iT - 37T^{2} \) |
| 41 | \( 1 - 7.94iT - 41T^{2} \) |
| 43 | \( 1 + 9.24T + 43T^{2} \) |
| 47 | \( 1 + 6.97iT - 47T^{2} \) |
| 53 | \( 1 - 6.11T + 53T^{2} \) |
| 59 | \( 1 - 12.1iT - 59T^{2} \) |
| 61 | \( 1 - 1.75T + 61T^{2} \) |
| 67 | \( 1 + 7.94iT - 67T^{2} \) |
| 71 | \( 1 + 0.735iT - 71T^{2} \) |
| 73 | \( 1 + 13.5iT - 73T^{2} \) |
| 79 | \( 1 + 5.66T + 79T^{2} \) |
| 83 | \( 1 - 13.7iT - 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 8.34iT - 97T^{2} \) |
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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
−11.38273254614174062813345216825, −10.22179324222868267366938361501, −8.870215906143924106040739041685, −8.408090681681962062337480590180, −7.68359736815646197965724295136, −6.37559923499779255873739449418, −5.40641223564695592750581498102, −4.85194328306705359141015143869, −3.11572100545275452388798437008, −0.939047223363470004914495963438,
2.06046572575067476436516906155, 2.92608673831503691028816730563, 3.78282394472996448841528630712, 5.66531183879857994908749316707, 6.64564015609238126559716533211, 7.40094339122801876023979053539, 8.828600845345598277908040188106, 9.844955990935650629798750531087, 10.58800923155299158985465261114, 11.32807571851903265089601824551