| L(s) = 1 | + 3-s + 4-s + 0.618·7-s + 9-s + 12-s − 1.61·13-s + 16-s − 1.61·19-s + 0.618·21-s + 27-s + 0.618·28-s + 0.618·31-s + 36-s − 1.61·37-s − 1.61·39-s − 1.61·43-s + 48-s − 0.618·49-s − 1.61·52-s − 1.61·57-s + 0.618·61-s + 0.618·63-s + 64-s + 0.618·67-s + 0.618·73-s − 1.61·76-s + 0.618·79-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s + 0.618·7-s + 9-s + 12-s − 1.61·13-s + 16-s − 1.61·19-s + 0.618·21-s + 27-s + 0.618·28-s + 0.618·31-s + 36-s − 1.61·37-s − 1.61·39-s − 1.61·43-s + 48-s − 0.618·49-s − 1.61·52-s − 1.61·57-s + 0.618·61-s + 0.618·63-s + 64-s + 0.618·67-s + 0.618·73-s − 1.61·76-s + 0.618·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.006313981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.006313981\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.61T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 37 | \( 1 + 1.61T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.618T + T^{2} \) |
| 79 | \( 1 - 0.618T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.446339304395252378756936285218, −8.352919072661964184498388481390, −8.017343415900767521055603353342, −7.01287969734905496239101128860, −6.64462712983853542535593412539, −5.26607733076567246515006120324, −4.45628745509455850602975363178, −3.34127458740237989358513250002, −2.36561830900024464891445528986, −1.78552479775240739760022477976,
1.78552479775240739760022477976, 2.36561830900024464891445528986, 3.34127458740237989358513250002, 4.45628745509455850602975363178, 5.26607733076567246515006120324, 6.64462712983853542535593412539, 7.01287969734905496239101128860, 8.017343415900767521055603353342, 8.352919072661964184498388481390, 9.446339304395252378756936285218
