| L(s) = 1 | − 0.618·2-s + 3-s − 1.61·4-s − 0.618·6-s − 0.618·7-s + 2.23·8-s + 9-s − 0.236·11-s − 1.61·12-s + 6.23·13-s + 0.381·14-s + 1.85·16-s + 6.61·17-s − 0.618·18-s − 5·19-s − 0.618·21-s + 0.145·22-s + 3.47·23-s + 2.23·24-s − 3.85·26-s + 27-s + 1.00·28-s + 6.70·29-s − 2.14·31-s − 5.61·32-s − 0.236·33-s − 4.09·34-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 0.577·3-s − 0.809·4-s − 0.252·6-s − 0.233·7-s + 0.790·8-s + 0.333·9-s − 0.0711·11-s − 0.467·12-s + 1.72·13-s + 0.102·14-s + 0.463·16-s + 1.60·17-s − 0.145·18-s − 1.14·19-s − 0.134·21-s + 0.0311·22-s + 0.723·23-s + 0.456·24-s − 0.755·26-s + 0.192·27-s + 0.188·28-s + 1.24·29-s − 0.385·31-s − 0.993·32-s − 0.0410·33-s − 0.701·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171102577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171102577\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 - 6.61T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 + 8.56T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 - 8.61T + 59T^{2} \) |
| 61 | \( 1 + 6.61T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 8.38T + 71T^{2} \) |
| 73 | \( 1 - 4.85T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 18.0T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−11.11345553373109061851264096755, −10.26155798475251351705298397436, −9.470975424975876191832367190146, −8.477965049286835379142242733742, −8.118559454885092348546072995490, −6.74341710917242514493546118349, −5.52409924291780871056368110165, −4.20465731223106558982676372221, −3.23884589731291312008518895743, −1.24922868285247279124293625879,
1.24922868285247279124293625879, 3.23884589731291312008518895743, 4.20465731223106558982676372221, 5.52409924291780871056368110165, 6.74341710917242514493546118349, 8.118559454885092348546072995490, 8.477965049286835379142242733742, 9.470975424975876191832367190146, 10.26155798475251351705298397436, 11.11345553373109061851264096755
