L(s) = 1 | + 2·2-s − 5.78·3-s + 4·4-s + 6.19·5-s − 11.5·6-s + 11.2·7-s + 8·8-s + 6.45·9-s + 12.3·10-s − 45.1·11-s − 23.1·12-s + 22.6·13-s + 22.4·14-s − 35.8·15-s + 16·16-s − 31.0·17-s + 12.9·18-s + 24.7·20-s − 64.9·21-s − 90.2·22-s − 144.·23-s − 46.2·24-s − 86.5·25-s + 45.3·26-s + 118.·27-s + 44.8·28-s + 24.2·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.11·3-s + 0.5·4-s + 0.554·5-s − 0.787·6-s + 0.605·7-s + 0.353·8-s + 0.238·9-s + 0.392·10-s − 1.23·11-s − 0.556·12-s + 0.483·13-s + 0.428·14-s − 0.617·15-s + 0.250·16-s − 0.442·17-s + 0.168·18-s + 0.277·20-s − 0.674·21-s − 0.874·22-s − 1.30·23-s − 0.393·24-s − 0.692·25-s + 0.341·26-s + 0.847·27-s + 0.302·28-s + 0.155·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 5.78T + 27T^{2} \) |
| 5 | \( 1 - 6.19T + 125T^{2} \) |
| 7 | \( 1 - 11.2T + 343T^{2} \) |
| 11 | \( 1 + 45.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.0T + 4.91e3T^{2} \) |
| 23 | \( 1 + 144.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 24.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 303.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 121.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 103.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 444.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 481.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 441.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 242.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 531.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 727.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 225.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 351.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 602.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 248.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.66e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 846.T + 9.12e5T^{2} \) |
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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
−10.09133564842582289687928095550, −8.547604128990597816323783369619, −7.74717930590740554736459708580, −6.49584707758645639499914935171, −5.88389821390692879701814521091, −5.14600474574963263525811320098, −4.36705200307461310774753661785, −2.83621882268981337765840736320, −1.63835454561042631940280827935, 0,
1.63835454561042631940280827935, 2.83621882268981337765840736320, 4.36705200307461310774753661785, 5.14600474574963263525811320098, 5.88389821390692879701814521091, 6.49584707758645639499914935171, 7.74717930590740554736459708580, 8.547604128990597816323783369619, 10.09133564842582289687928095550