L(s) = 1 | − 1.90·2-s + 2.61·4-s − 1.61·7-s − 3.07·8-s − 0.618·13-s + 3.07·14-s + 3.23·16-s + 1.17·17-s + 0.618·19-s + 1.90·23-s + 25-s + 1.17·26-s − 4.23·28-s − 1.17·29-s − 3.07·32-s − 2.23·34-s − 1.17·38-s + 1.90·41-s − 3.61·46-s + 1.61·49-s − 1.90·50-s − 1.61·52-s + 4.97·56-s + 2.23·58-s − 1.17·59-s + 1.61·61-s + 2.61·64-s + ⋯ |
L(s) = 1 | − 1.90·2-s + 2.61·4-s − 1.61·7-s − 3.07·8-s − 0.618·13-s + 3.07·14-s + 3.23·16-s + 1.17·17-s + 0.618·19-s + 1.90·23-s + 25-s + 1.17·26-s − 4.23·28-s − 1.17·29-s − 3.07·32-s − 2.23·34-s − 1.17·38-s + 1.90·41-s − 3.61·46-s + 1.61·49-s − 1.90·50-s − 1.61·52-s + 4.97·56-s + 2.23·58-s − 1.17·59-s + 1.61·61-s + 2.61·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3670410436\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3670410436\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 1.90T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 0.618T + T^{2} \) |
| 17 | \( 1 - 1.17T + T^{2} \) |
| 19 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 - 1.90T + T^{2} \) |
| 29 | \( 1 + 1.17T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.90T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.17T + T^{2} \) |
| 61 | \( 1 - 1.61T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 0.618T + T^{2} \) |
| 83 | \( 1 - 1.17T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.61T + T^{2} \) |
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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
−10.02087273059687752694258191953, −9.333717582234637085570573417833, −9.014249176300606203703455736591, −7.70684529459321721397934212923, −7.18551444495452227673406772615, −6.43069731405788428049546904375, −5.44016798597709154893533782543, −3.35135815146689238618968971753, −2.65158796232566032746354303934, −0.939426999057368386969181400032,
0.939426999057368386969181400032, 2.65158796232566032746354303934, 3.35135815146689238618968971753, 5.44016798597709154893533782543, 6.43069731405788428049546904375, 7.18551444495452227673406772615, 7.70684529459321721397934212923, 9.014249176300606203703455736591, 9.333717582234637085570573417833, 10.02087273059687752694258191953