L(s) = 1 | + (−0.939 + 0.342i)2-s + (1.72 + 0.627i)3-s + (0.766 − 0.642i)4-s + (−0.326 − 1.85i)5-s − 1.83·6-s + (0.598 + 3.39i)7-s + (−0.500 + 0.866i)8-s + (0.278 + 0.233i)9-s + (0.939 + 1.62i)10-s + (1.40 − 2.43i)11-s + (1.72 − 0.627i)12-s + (−2.65 + 2.23i)13-s + (−1.72 − 2.98i)14-s + (0.598 − 3.39i)15-s + (0.173 − 0.984i)16-s + (−2.37 − 1.99i)17-s + ⋯ |
L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.994 + 0.362i)3-s + (0.383 − 0.321i)4-s + (−0.145 − 0.827i)5-s − 0.748·6-s + (0.226 + 1.28i)7-s + (−0.176 + 0.306i)8-s + (0.0927 + 0.0777i)9-s + (0.297 + 0.514i)10-s + (0.423 − 0.733i)11-s + (0.497 − 0.181i)12-s + (−0.737 + 0.618i)13-s + (−0.460 − 0.797i)14-s + (0.154 − 0.876i)15-s + (0.0434 − 0.246i)16-s + (−0.575 − 0.483i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.870674 + 0.167383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.870674 + 0.167383i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 37 | \( 1 + (2.56 - 5.51i)T \) |
good | 3 | \( 1 + (-1.72 - 0.627i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.326 + 1.85i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.598 - 3.39i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.40 + 2.43i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.65 - 2.23i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.37 + 1.99i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (6.99 + 2.54i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.321 - 0.557i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.08 - 1.87i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.90T + 31T^{2} \) |
| 41 | \( 1 + (8.13 - 6.82i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 - 8.30T + 43T^{2} \) |
| 47 | \( 1 + (-3.92 - 6.80i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.839 + 4.76i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.0961 + 0.545i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.37 + 4.50i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.0366 - 0.207i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.75 + 1.72i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + (0.330 + 1.87i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-5.21 - 4.37i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (1.68 - 9.54i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (8.52 + 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.93800715539082527281357025172, −13.83495500100852360719255545386, −12.35536288540429063027082934257, −11.35739359984355807894208009172, −9.610186624143061983498738984193, −8.724649244667358222488303155745, −8.417437348373399069943889553352, −6.43193263559365270436746016883, −4.71995734231250318682238379607, −2.54006356054050893070693495172,
2.31994716054009641416243844545, 3.98282362841921112276334608742, 6.81210004841551863407061843341, 7.61627413552245467125509416649, 8.659827450074097166176740484543, 10.19770910535915263496537469185, 10.79591377197980021092610537142, 12.37259948542458278860797179981, 13.56049440175055657754565301191, 14.57163821606939409590157019002