L(s) = 1 | + 0.792i·2-s − 2.52i·3-s + 1.37·4-s + (−2.18 − 0.469i)5-s + 2·6-s + 3.46i·7-s + 2.67i·8-s − 3.37·9-s + (0.372 − 1.73i)10-s − 11-s − 3.46i·12-s − 2.74·14-s + (−1.18 + 5.51i)15-s + 0.627·16-s − 5.04i·17-s − 2.67i·18-s + ⋯ |
L(s) = 1 | + 0.560i·2-s − 1.45i·3-s + 0.686·4-s + (−0.977 − 0.210i)5-s + 0.816·6-s + 1.30i·7-s + 0.944i·8-s − 1.12·9-s + (0.117 − 0.547i)10-s − 0.301·11-s − 0.999i·12-s − 0.733·14-s + (−0.306 + 1.42i)15-s + 0.156·16-s − 1.22i·17-s − 0.629i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.860315 - 0.0914128i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.860315 - 0.0914128i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.18 + 0.469i)T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 0.792iT - 2T^{2} \) |
| 3 | \( 1 + 2.52iT - 3T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 5.04iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 2.52iT - 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 31 | \( 1 + 2.37T + 31T^{2} \) |
| 37 | \( 1 + 11.0iT - 37T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 6.63iT - 47T^{2} \) |
| 53 | \( 1 - 3.16iT - 53T^{2} \) |
| 59 | \( 1 - 1.62T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 0.644iT - 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 6.63iT - 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 + 4.10iT - 97T^{2} \) |
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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
−15.45163900919693144083861587179, −14.32625908956387078472543558768, −12.77160542675294909516894765829, −12.04501127844926345527565061308, −11.23905208913192929239537573227, −8.752358483478454298537765325143, −7.75464132179680785468636652988, −6.85437210822826851515912330194, −5.52923801730463138407263999870, −2.45217711766523776713592093680,
3.47984059806640897081261858555, 4.38975127802683186965272439597, 6.76134604263451556055217846691, 8.261424559393711865296506807208, 10.21784533693697580003851458916, 10.55003680888481659010168730285, 11.51323317728160153252196799477, 12.92369122918368406875878966078, 14.69536127088178497001264221151, 15.39752858295455827622015830390