| L(s) = 1 | − 1.61·2-s + 1.61·4-s − 0.618·7-s − 8-s + 11-s − 13-s + 1.00·14-s + 1.61·19-s − 1.61·22-s − 0.618·23-s + 25-s + 1.61·26-s − 1.00·28-s + 32-s − 2.61·38-s + 0.618·41-s + 1.61·44-s + 1.00·46-s − 0.618·49-s − 1.61·50-s − 1.61·52-s + 1.61·53-s + 0.618·56-s − 1.61·64-s + 1.61·73-s + 2.61·76-s − 0.618·77-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.61·4-s − 0.618·7-s − 8-s + 11-s − 13-s + 1.00·14-s + 1.61·19-s − 1.61·22-s − 0.618·23-s + 25-s + 1.61·26-s − 1.00·28-s + 32-s − 2.61·38-s + 0.618·41-s + 1.61·44-s + 1.00·46-s − 0.618·49-s − 1.61·50-s − 1.61·52-s + 1.61·53-s + 0.618·56-s − 1.61·64-s + 1.61·73-s + 2.61·76-s − 0.618·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4787380897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4787380897\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 0.618T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.61T + T^{2} \) |
| 23 | \( 1 + 0.618T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.61T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.618T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.673488512536933188945348450431, −9.240446081369098313325319121893, −8.406783343682521185060621690191, −7.44280174016165141610831880825, −6.97916596973920122118746110781, −6.06259160672395650214261379153, −4.82552517804463552985268493104, −3.47273873382208245584091634872, −2.32766453594405257828719345240, −0.985901173479787071132460001469,
0.985901173479787071132460001469, 2.32766453594405257828719345240, 3.47273873382208245584091634872, 4.82552517804463552985268493104, 6.06259160672395650214261379153, 6.97916596973920122118746110781, 7.44280174016165141610831880825, 8.406783343682521185060621690191, 9.240446081369098313325319121893, 9.673488512536933188945348450431
