| L(s) = 1 | + 2-s + 4-s + 0.618·7-s + 8-s + 9-s − 1.61·11-s − 1.61·13-s + 0.618·14-s + 16-s + 18-s + 0.618·19-s − 1.61·22-s − 1.61·23-s − 1.61·26-s + 0.618·28-s + 32-s + 36-s + 0.618·37-s + 0.618·38-s + 0.618·41-s − 1.61·44-s − 1.61·46-s − 1.61·47-s − 0.618·49-s − 1.61·52-s + 0.618·53-s + 0.618·56-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 0.618·7-s + 8-s + 9-s − 1.61·11-s − 1.61·13-s + 0.618·14-s + 16-s + 18-s + 0.618·19-s − 1.61·22-s − 1.61·23-s − 1.61·26-s + 0.618·28-s + 32-s + 36-s + 0.618·37-s + 0.618·38-s + 0.618·41-s − 1.61·44-s − 1.61·46-s − 1.61·47-s − 0.618·49-s − 1.61·52-s + 0.618·53-s + 0.618·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.884194617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.884194617\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 + 1.61T + T^{2} \) |
| 13 | \( 1 + 1.61T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 + 1.61T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.61T + T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 - 0.618T + T^{2} \) |
| 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 | \( 1 + 1.61T + T^{2} \) |
| 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
−10.12447154852178158358190119171, −9.824363199114779905367420181779, −7.974849957638779673672109357413, −7.69793171148142051841559900022, −6.81047264717712644387283226745, −5.57094684763527530974637488804, −4.93014192469695967243105720168, −4.19102068368152253136825469476, −2.80912449030115518576986234228, −1.91840511529439621330329674315,
1.91840511529439621330329674315, 2.80912449030115518576986234228, 4.19102068368152253136825469476, 4.93014192469695967243105720168, 5.57094684763527530974637488804, 6.81047264717712644387283226745, 7.69793171148142051841559900022, 7.974849957638779673672109357413, 9.824363199114779905367420181779, 10.12447154852178158358190119171
