| L(s) = 1 | + 2-s + 4-s + 8-s + 9-s − 1.61·13-s + 16-s + 0.618·17-s + 18-s − 1.61·26-s + 0.618·29-s + 32-s + 0.618·34-s + 36-s + 0.618·37-s − 1.61·41-s + 49-s − 1.61·52-s − 1.61·53-s + 0.618·58-s − 1.61·61-s + 64-s + 0.618·68-s + 72-s − 1.61·73-s + 0.618·74-s + 81-s − 1.61·82-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 8-s + 9-s − 1.61·13-s + 16-s + 0.618·17-s + 18-s − 1.61·26-s + 0.618·29-s + 32-s + 0.618·34-s + 36-s + 0.618·37-s − 1.61·41-s + 49-s − 1.61·52-s − 1.61·53-s + 0.618·58-s − 1.61·61-s + 64-s + 0.618·68-s + 72-s − 1.61·73-s + 0.618·74-s + 81-s − 1.61·82-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.407807284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.407807284\) |
| \(L(1)\) |
|
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 | \( 1 + 1.61T + T^{2} \) |
| 17 | \( 1 - 0.618T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 0.618T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 97 | \( 1 - 0.618T + T^{2} \) |
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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
−9.262307737681312760776599112905, −7.987903483519576702865248525016, −7.41381432716445210967015622777, −6.79749685349730115313334974778, −5.93052022183563601459213816861, −4.92339263283094638127634403249, −4.54351524441054160044893494989, −3.46670385188677177919412671745, −2.56568903812152465160585577058, −1.51324887502618263691737617306,
1.51324887502618263691737617306, 2.56568903812152465160585577058, 3.46670385188677177919412671745, 4.54351524441054160044893494989, 4.92339263283094638127634403249, 5.93052022183563601459213816861, 6.79749685349730115313334974778, 7.41381432716445210967015622777, 7.987903483519576702865248525016, 9.262307737681312760776599112905
