L(s) = 1 | − 1.49·2-s − 2.14·3-s + 0.247·4-s + 3.21·6-s − 0.914·7-s + 2.62·8-s + 1.59·9-s − 5.69·11-s − 0.529·12-s + 5.13·13-s + 1.37·14-s − 4.43·16-s − 5.54·17-s − 2.38·18-s + 4.24·19-s + 1.96·21-s + 8.53·22-s − 7.15·23-s − 5.63·24-s − 7.69·26-s + 3.01·27-s − 0.226·28-s + 1.00·29-s − 6.52·31-s + 1.39·32-s + 12.1·33-s + 8.31·34-s + ⋯ |
L(s) = 1 | − 1.05·2-s − 1.23·3-s + 0.123·4-s + 1.31·6-s − 0.345·7-s + 0.929·8-s + 0.530·9-s − 1.71·11-s − 0.152·12-s + 1.42·13-s + 0.366·14-s − 1.10·16-s − 1.34·17-s − 0.562·18-s + 0.974·19-s + 0.427·21-s + 1.81·22-s − 1.49·23-s − 1.14·24-s − 1.50·26-s + 0.580·27-s − 0.0427·28-s + 0.186·29-s − 1.17·31-s + 0.245·32-s + 2.12·33-s + 1.42·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2502865877\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2502865877\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + 1.49T + 2T^{2} \) |
| 3 | \( 1 + 2.14T + 3T^{2} \) |
| 7 | \( 1 + 0.914T + 7T^{2} \) |
| 11 | \( 1 + 5.69T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 + 5.54T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 7.15T + 23T^{2} \) |
| 29 | \( 1 - 1.00T + 29T^{2} \) |
| 31 | \( 1 + 6.52T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 4.42T + 41T^{2} \) |
| 43 | \( 1 + 0.310T + 43T^{2} \) |
| 53 | \( 1 - 6.38T + 53T^{2} \) |
| 59 | \( 1 - 1.42T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 4.25T + 67T^{2} \) |
| 71 | \( 1 - 3.83T + 71T^{2} \) |
| 73 | \( 1 - 9.86T + 73T^{2} \) |
| 79 | \( 1 + 6.00T + 79T^{2} \) |
| 83 | \( 1 + 2.43T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−10.01812546549299511806759288492, −8.873355167932020469138694147746, −8.305259488104345755798889610661, −7.37554309973644121213861407966, −6.49124668649865829614857875558, −5.57023449698490132887949616352, −4.90022513342893019534585390855, −3.63770060260820964883859344461, −1.98650487601758783984399315435, −0.45441100903460676846008652280,
0.45441100903460676846008652280, 1.98650487601758783984399315435, 3.63770060260820964883859344461, 4.90022513342893019534585390855, 5.57023449698490132887949616352, 6.49124668649865829614857875558, 7.37554309973644121213861407966, 8.305259488104345755798889610661, 8.873355167932020469138694147746, 10.01812546549299511806759288492