| L(s) = 1 | + 3-s + 4-s − 1.61·7-s + 9-s + 12-s + 0.618·13-s + 16-s + 0.618·19-s − 1.61·21-s + 25-s + 27-s − 1.61·28-s − 1.61·31-s + 36-s + 0.618·37-s + 0.618·39-s − 1.61·43-s + 48-s + 1.61·49-s + 0.618·52-s + 0.618·57-s − 1.61·61-s − 1.61·63-s + 64-s + 0.618·67-s + 0.618·73-s + 75-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s − 1.61·7-s + 9-s + 12-s + 0.618·13-s + 16-s + 0.618·19-s − 1.61·21-s + 25-s + 27-s − 1.61·28-s − 1.61·31-s + 36-s + 0.618·37-s + 0.618·39-s − 1.61·43-s + 48-s + 1.61·49-s + 0.618·52-s + 0.618·57-s − 1.61·61-s − 1.61·63-s + 64-s + 0.618·67-s + 0.618·73-s + 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.949021709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.949021709\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 0.618T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.61T + T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 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 | \( 1 + 1.61T + T^{2} \) |
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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
−9.212567410568116348548608903685, −8.319369742445449833421992563354, −7.48147756599130034164163348478, −6.81032615497590136863509178488, −6.32338790889333661660590147388, −5.29872531455475599729434714340, −3.85051768886993216245044059385, −3.25969146970746027755595869873, −2.63032625212897928134545401334, −1.43607811426819791461902103641,
1.43607811426819791461902103641, 2.63032625212897928134545401334, 3.25969146970746027755595869873, 3.85051768886993216245044059385, 5.29872531455475599729434714340, 6.32338790889333661660590147388, 6.81032615497590136863509178488, 7.48147756599130034164163348478, 8.319369742445449833421992563354, 9.212567410568116348548608903685
