L(s) = 1 | + 2-s + 0.763·3-s + 4-s − 5-s + 0.763·6-s − 4.04·7-s + 8-s − 2.41·9-s − 10-s + 0.263·11-s + 0.763·12-s + 3.03·13-s − 4.04·14-s − 0.763·15-s + 16-s − 0.160·17-s − 2.41·18-s + 1.01·19-s − 20-s − 3.08·21-s + 0.263·22-s + 0.763·24-s + 25-s + 3.03·26-s − 4.13·27-s − 4.04·28-s + 5.02·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.440·3-s + 0.5·4-s − 0.447·5-s + 0.311·6-s − 1.52·7-s + 0.353·8-s − 0.805·9-s − 0.316·10-s + 0.0793·11-s + 0.220·12-s + 0.842·13-s − 1.08·14-s − 0.197·15-s + 0.250·16-s − 0.0389·17-s − 0.569·18-s + 0.233·19-s − 0.223·20-s − 0.673·21-s + 0.0561·22-s + 0.155·24-s + 0.200·25-s + 0.595·26-s − 0.795·27-s − 0.764·28-s + 0.932·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.459394073\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.459394073\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 0.763T + 3T^{2} \) |
| 7 | \( 1 + 4.04T + 7T^{2} \) |
| 11 | \( 1 - 0.263T + 11T^{2} \) |
| 13 | \( 1 - 3.03T + 13T^{2} \) |
| 17 | \( 1 + 0.160T + 17T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 29 | \( 1 - 5.02T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 - 4.05T + 41T^{2} \) |
| 43 | \( 1 + 0.864T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 1.97T + 59T^{2} \) |
| 61 | \( 1 - 4.65T + 61T^{2} \) |
| 67 | \( 1 + 3.68T + 67T^{2} \) |
| 71 | \( 1 + 3.70T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 1.31T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.098357169407422933305195954597, −7.45224083751117212946804917054, −6.51022881807154616418958646302, −6.15251648753162138598658153134, −5.36971975123429321459405151781, −4.30141252751415869335213776890, −3.53224334804277584267242027138, −3.11403106764569521210370983195, −2.28592752727523767795302145345, −0.71879976502141093745053369015,
0.71879976502141093745053369015, 2.28592752727523767795302145345, 3.11403106764569521210370983195, 3.53224334804277584267242027138, 4.30141252751415869335213776890, 5.36971975123429321459405151781, 6.15251648753162138598658153134, 6.51022881807154616418958646302, 7.45224083751117212946804917054, 8.098357169407422933305195954597