| L(s) = 1 | + 1.61·2-s − 3-s + 0.618·4-s − 1.61·6-s − 0.618·7-s − 2.23·8-s − 2·9-s − 5.23·11-s − 0.618·12-s − 1.85·13-s − 1.00·14-s − 4.85·16-s + 5.23·17-s − 3.23·18-s + 0.854·19-s + 0.618·21-s − 8.47·22-s − 3.76·23-s + 2.23·24-s − 3·26-s + 5·27-s − 0.381·28-s − 3.61·29-s − 3·31-s − 3.38·32-s + 5.23·33-s + 8.47·34-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.660·6-s − 0.233·7-s − 0.790·8-s − 0.666·9-s − 1.57·11-s − 0.178·12-s − 0.514·13-s − 0.267·14-s − 1.21·16-s + 1.26·17-s − 0.762·18-s + 0.195·19-s + 0.134·21-s − 1.80·22-s − 0.784·23-s + 0.456·24-s − 0.588·26-s + 0.962·27-s − 0.0721·28-s − 0.671·29-s − 0.538·31-s − 0.597·32-s + 0.911·33-s + 1.45·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 + 0.618T + 47T^{2} \) |
| 53 | \( 1 - 3.47T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 + 6.61T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 8.09T + 79T^{2} \) |
| 83 | \( 1 - 6.23T + 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−10.33974219662881949813925812997, −9.472106703208809998437932716325, −8.271859739401738772609051754587, −7.37965171227660629349863866049, −6.03946997729463624803435756299, −5.50919505674424159480558476977, −4.82603976515769184310932348181, −3.49313837703386866870810729024, −2.58187502389972903250512989258, 0,
2.58187502389972903250512989258, 3.49313837703386866870810729024, 4.82603976515769184310932348181, 5.50919505674424159480558476977, 6.03946997729463624803435756299, 7.37965171227660629349863866049, 8.271859739401738772609051754587, 9.472106703208809998437932716325, 10.33974219662881949813925812997
