| L(s) = 1 | + 0.618·2-s − 2.61·3-s − 1.61·4-s − 1.61·6-s − 3·7-s − 2.23·8-s + 3.85·9-s − 3·11-s + 4.23·12-s + 1.85·13-s − 1.85·14-s + 1.85·16-s − 0.236·17-s + 2.38·18-s − 1.38·19-s + 7.85·21-s − 1.85·22-s + 3.23·23-s + 5.85·24-s + 1.14·26-s − 2.23·27-s + 4.85·28-s − 6.70·29-s − 6.09·31-s + 5.61·32-s + 7.85·33-s − 0.145·34-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 1.51·3-s − 0.809·4-s − 0.660·6-s − 1.13·7-s − 0.790·8-s + 1.28·9-s − 0.904·11-s + 1.22·12-s + 0.514·13-s − 0.495·14-s + 0.463·16-s − 0.0572·17-s + 0.561·18-s − 0.317·19-s + 1.71·21-s − 0.395·22-s + 0.674·23-s + 1.19·24-s + 0.224·26-s − 0.430·27-s + 0.917·28-s − 1.24·29-s − 1.09·31-s + 0.993·32-s + 1.36·33-s − 0.0250·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2.61T + 3T^{2} \) |
| 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
−12.93972281835560793328304327272, −12.05382558703418650096099764436, −10.86172981070757279343267880448, −9.986551135646300424046204636967, −8.780613974808470968139799786105, −6.98778454542719658500051129123, −5.83997881754905965570583860943, −5.10791352937727896511830478381, −3.59379870547119281719423165809, 0,
3.59379870547119281719423165809, 5.10791352937727896511830478381, 5.83997881754905965570583860943, 6.98778454542719658500051129123, 8.780613974808470968139799786105, 9.986551135646300424046204636967, 10.86172981070757279343267880448, 12.05382558703418650096099764436, 12.93972281835560793328304327272
