L(s) = 1 | + (−0.5 − 0.866i)2-s + (1 + 1.73i)3-s + (−0.499 + 0.866i)4-s + (0.999 − 1.73i)6-s + 2·7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + 11-s − 1.99·12-s + (0.5 − 0.866i)13-s + (−1 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (3 + 5.19i)17-s + 0.999·18-s + (−3.5 + 2.59i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.577 + 0.999i)3-s + (−0.249 + 0.433i)4-s + (0.408 − 0.707i)6-s + 0.755·7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + 0.301·11-s − 0.577·12-s + (0.138 − 0.240i)13-s + (−0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.727 + 1.26i)17-s + 0.235·18-s + (−0.802 + 0.596i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46406 + 0.316424i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46406 + 0.316424i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + (3.5 - 2.59i)T \) |
good | 3 | \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.5 + 7.79i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.88001311492037414070644940726, −10.40060935443841962456662611276, −9.600714985020813527948544248553, −8.467779840250957616275657126665, −8.234048812911613774936302853707, −6.62770548705313872563619376033, −5.13831236350976290999180483901, −4.08903152934446920283427507627, −3.27876004991778180731223911297, −1.67531807233687475215544391971,
1.24521425287284537056541760796, 2.59351915792078184431253759349, 4.40904864180515439500091636305, 5.56151146100428010617863353430, 6.84542093230910216464929138403, 7.40561654795960395919066679000, 8.285603398092119200046154165400, 8.976529768976431548031177304582, 10.04888732771626787755320277732, 11.22682234217004666619911428810