| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s − 0.618·13-s + 16-s + 1.61·17-s − 18-s + 0.618·26-s − 1.61·29-s − 32-s − 1.61·34-s + 36-s + 1.61·37-s + 0.618·41-s + 49-s − 0.618·52-s − 0.618·53-s + 1.61·58-s + 0.618·61-s + 64-s + 1.61·68-s − 72-s − 0.618·73-s − 1.61·74-s + 81-s − 0.618·82-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s − 0.618·13-s + 16-s + 1.61·17-s − 18-s + 0.618·26-s − 1.61·29-s − 32-s − 1.61·34-s + 36-s + 1.61·37-s + 0.618·41-s + 49-s − 0.618·52-s − 0.618·53-s + 1.61·58-s + 0.618·61-s + 64-s + 1.61·68-s − 72-s − 0.618·73-s − 1.61·74-s + 81-s − 0.618·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8599913799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8599913799\) |
| \(L(1)\) |
|
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 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 0.618T + T^{2} \) |
| 17 | \( 1 - 1.61T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.61T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.61T + T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 0.618T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 - 1.61T + T^{2} \) |
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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
−9.350565196064229383003616765204, −8.261927763184136426272610755036, −7.50358998789650268085332253260, −7.23524900688945338908853768260, −6.11307589142641868293019243987, −5.41644235033860386479727801822, −4.21411698516904211507860939868, −3.22031490164283915784657848588, −2.13213772147647055134379176861, −1.06887600902116701933945795472,
1.06887600902116701933945795472, 2.13213772147647055134379176861, 3.22031490164283915784657848588, 4.21411698516904211507860939868, 5.41644235033860386479727801822, 6.11307589142641868293019243987, 7.23524900688945338908853768260, 7.50358998789650268085332253260, 8.261927763184136426272610755036, 9.350565196064229383003616765204
