L(s) = 1 | − 0.618·2-s − 3-s − 1.61·4-s + 0.618·6-s + 1.61·7-s + 2.23·8-s − 2·9-s − 0.763·11-s + 1.61·12-s + 4.85·13-s − 1.00·14-s + 1.85·16-s + 0.763·17-s + 1.23·18-s − 5.85·19-s − 1.61·21-s + 0.472·22-s − 8.23·23-s − 2.23·24-s − 3.00·26-s + 5·27-s − 2.61·28-s − 1.38·29-s − 3·31-s − 5.61·32-s + 0.763·33-s − 0.472·34-s + ⋯ |
L(s) = 1 | − 0.437·2-s − 0.577·3-s − 0.809·4-s + 0.252·6-s + 0.611·7-s + 0.790·8-s − 0.666·9-s − 0.230·11-s + 0.467·12-s + 1.34·13-s − 0.267·14-s + 0.463·16-s + 0.185·17-s + 0.291·18-s − 1.34·19-s − 0.353·21-s + 0.100·22-s − 1.71·23-s − 0.456·24-s − 0.588·26-s + 0.962·27-s − 0.494·28-s − 0.256·29-s − 0.538·31-s − 0.993·32-s + 0.132·33-s − 0.0809·34-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 | \( 1 + 0.618T + 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 - 1.61T + 47T^{2} \) |
| 53 | \( 1 + 5.47T + 53T^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 + 9.23T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 - 3.09T + 79T^{2} \) |
| 83 | \( 1 - 1.76T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 + 2.85T + 97T^{2} \) |
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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
−10.37836332430107856791355928771, −9.158769433767514207650242644238, −8.402407948670096481927558824071, −7.911494512284177911245429121248, −6.39846032466690188750117531412, −5.61936346762485966861314235998, −4.63158370589134938107945374811, −3.61982368548635112315187805330, −1.70627417996651129056379359719, 0,
1.70627417996651129056379359719, 3.61982368548635112315187805330, 4.63158370589134938107945374811, 5.61936346762485966861314235998, 6.39846032466690188750117531412, 7.911494512284177911245429121248, 8.402407948670096481927558824071, 9.158769433767514207650242644238, 10.37836332430107856791355928771