| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 1.56·7-s + 8-s + 9-s + 10-s − 1.56·11-s − 12-s + 2·13-s − 1.56·14-s − 15-s + 16-s + 5.12·17-s + 18-s + 4.68·19-s + 20-s + 1.56·21-s − 1.56·22-s + 5.56·23-s − 24-s + 25-s + 2·26-s − 27-s − 1.56·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.590·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.470·11-s − 0.288·12-s + 0.554·13-s − 0.417·14-s − 0.258·15-s + 0.250·16-s + 1.24·17-s + 0.235·18-s + 1.07·19-s + 0.223·20-s + 0.340·21-s − 0.332·22-s + 1.15·23-s − 0.204·24-s + 0.200·25-s + 0.392·26-s − 0.192·27-s − 0.295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.199795855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.199795855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 + 7.80T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 4.87T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 9.36T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 - 9.80T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 2.24T + 83T^{2} \) |
| 89 | \( 1 + 1.31T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.07896814888613568901950952352, −9.560176306576161195048511404639, −8.269220544903797731966262923998, −7.25349638059761408898454286542, −6.48159766284164875326425405871, −5.56865023429522812415694669481, −5.06919548701767731828346583917, −3.69405827722768577241665985395, −2.82179157383162195127400833664, −1.19219428858697637835555291853,
1.19219428858697637835555291853, 2.82179157383162195127400833664, 3.69405827722768577241665985395, 5.06919548701767731828346583917, 5.56865023429522812415694669481, 6.48159766284164875326425405871, 7.25349638059761408898454286542, 8.269220544903797731966262923998, 9.560176306576161195048511404639, 10.07896814888613568901950952352
