L(s) = 1 | − 1.61·7-s + 9-s − 0.618·11-s − 0.618·13-s + 1.61·19-s + 0.618·23-s + 1.61·37-s − 1.61·41-s + 0.618·47-s + 1.61·49-s + 1.61·53-s + 1.61·59-s − 1.61·63-s + 1.00·77-s + 81-s + 0.618·89-s + 1.00·91-s − 0.618·99-s + 0.618·103-s − 0.618·117-s + ⋯ |
L(s) = 1 | − 1.61·7-s + 9-s − 0.618·11-s − 0.618·13-s + 1.61·19-s + 0.618·23-s + 1.61·37-s − 1.61·41-s + 0.618·47-s + 1.61·49-s + 1.61·53-s + 1.61·59-s − 1.61·63-s + 1.00·77-s + 81-s + 0.618·89-s + 1.00·91-s − 0.618·99-s + 0.618·103-s − 0.618·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.102235948\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.102235948\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 + 0.618T + T^{2} \) |
| 13 | \( 1 + 0.618T + T^{2} \) |
| 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 | \( 1 - 1.61T + T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.618T + T^{2} \) |
| 53 | \( 1 - 1.61T + T^{2} \) |
| 59 | \( 1 - 1.61T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 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
−8.752024304406931343688087219878, −7.64796911616419341403190783602, −7.17207274093829513784128685220, −6.57440638480371128115291869599, −5.63919416550180467860781584161, −4.95219151167126215652308984753, −3.92073907315992392663511314950, −3.17289441575063098489022149902, −2.39427068704309268499552587773, −0.887985730505008628732700649258,
0.887985730505008628732700649258, 2.39427068704309268499552587773, 3.17289441575063098489022149902, 3.92073907315992392663511314950, 4.95219151167126215652308984753, 5.63919416550180467860781584161, 6.57440638480371128115291869599, 7.17207274093829513784128685220, 7.64796911616419341403190783602, 8.752024304406931343688087219878