| L(s) = 1 | − 0.618·2-s − 1.61·4-s − 3·7-s + 2.23·8-s + 3·11-s + 1.85·13-s + 1.85·14-s + 1.85·16-s + 0.236·17-s − 1.38·19-s − 1.85·22-s − 3.23·23-s − 1.14·26-s + 4.85·28-s + 6.70·29-s − 6.09·31-s − 5.61·32-s − 0.145·34-s − 9.70·37-s + 0.854·38-s + 3·41-s − 9·43-s − 4.85·44-s + 2.00·46-s − 7.32·47-s + 2·49-s − 3·52-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s − 1.13·7-s + 0.790·8-s + 0.904·11-s + 0.514·13-s + 0.495·14-s + 0.463·16-s + 0.0572·17-s − 0.317·19-s − 0.395·22-s − 0.674·23-s − 0.224·26-s + 0.917·28-s + 1.24·29-s − 1.09·31-s − 0.993·32-s − 0.0250·34-s − 1.59·37-s + 0.138·38-s + 0.468·41-s − 1.37·43-s − 0.731·44-s + 0.294·46-s − 1.06·47-s + 0.285·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 - 0.236T + 17T^{2} \) |
| 19 | \( 1 + 1.38T + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 6.09T + 31T^{2} \) |
| 37 | \( 1 + 9.70T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 9T + 43T^{2} \) |
| 47 | \( 1 + 7.32T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 5.09T + 61T^{2} \) |
| 67 | \( 1 + 7.14T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + 4.85T + 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 1.14T + 97T^{2} \) |
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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
−9.364548413281520664964006656047, −8.767499683874449884153001477285, −7.982644500631273399449482634357, −6.83618840359182586655778819742, −6.20349528195766851787401906927, −5.06964172214320127073450208450, −3.99124950656939240773113677567, −3.28553482112215943120812418959, −1.53511278705354164363266331654, 0,
1.53511278705354164363266331654, 3.28553482112215943120812418959, 3.99124950656939240773113677567, 5.06964172214320127073450208450, 6.20349528195766851787401906927, 6.83618840359182586655778819742, 7.982644500631273399449482634357, 8.767499683874449884153001477285, 9.364548413281520664964006656047
