L(s) = 1 | − 1.65·2-s − 1.73·3-s − 1.27·4-s + 4.21i·5-s + 2.85·6-s + 2.64i·7-s + 8.70·8-s + 2.99·9-s − 6.95i·10-s − 4.54i·11-s + 2.20·12-s + 7.59·13-s − 4.36i·14-s − 7.30i·15-s − 9.27·16-s + 3.10i·17-s + ⋯ |
L(s) = 1 | − 0.825·2-s − 0.577·3-s − 0.318·4-s + 0.842i·5-s + 0.476·6-s + 0.377i·7-s + 1.08·8-s + 0.333·9-s − 0.695i·10-s − 0.413i·11-s + 0.183·12-s + 0.584·13-s − 0.311i·14-s − 0.486i·15-s − 0.579·16-s + 0.182i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6414260592\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6414260592\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73T \) |
| 7 | \( 1 - 2.64iT \) |
| 23 | \( 1 + (-22.5 + 4.76i)T \) |
good | 2 | \( 1 + 1.65T + 4T^{2} \) |
| 5 | \( 1 - 4.21iT - 25T^{2} \) |
| 11 | \( 1 + 4.54iT - 121T^{2} \) |
| 13 | \( 1 - 7.59T + 169T^{2} \) |
| 17 | \( 1 - 3.10iT - 289T^{2} \) |
| 19 | \( 1 + 12.4iT - 361T^{2} \) |
| 29 | \( 1 + 6.35T + 841T^{2} \) |
| 31 | \( 1 + 30.1T + 961T^{2} \) |
| 37 | \( 1 - 62.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 58.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 46.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 7.95T + 2.20e3T^{2} \) |
| 53 | \( 1 + 24.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 32.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 81.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 100. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 40.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 57.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 116. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 38.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 21.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 68.3iT - 9.40e3T^{2} \) |
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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
−10.96598861377410454307074079443, −10.19013049739321135177642513517, −9.218836302600677550628947313083, −8.494339126773544507075743263339, −7.40745418570095640936319547424, −6.56125784095210724800149161774, −5.48173728342090660597446999685, −4.33157297209722332967132470953, −2.91643164523290732661384197775, −1.14794088142912331101507904417,
0.47599227723263116278878694259, 1.58756569080200640775877657211, 3.86652572524779829110208632993, 4.80193414191357007903786491101, 5.71294474688762840021841113234, 7.12299492602029788851007062920, 7.86086154009161594475498021436, 8.994984891729255466145220985792, 9.384538548222451401479946619865, 10.56757518921617087479471564365