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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.558581974\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.558581974\) |
\(L(1)\) |
|
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
−10.33649315983170256938586077098, −9.855559264285757365894589158273, −8.460558806753381159575475572854, −7.09027976863346583663906770029, −6.65956504988089730504825770078, −5.89831921974029083967596286995, −4.92433245517473811817350748300, −3.98788792494418015169251516367, −3.13738861457169047996351408440, −2.27500541487441074042644019185,
2.27500541487441074042644019185, 3.13738861457169047996351408440, 3.98788792494418015169251516367, 4.92433245517473811817350748300, 5.89831921974029083967596286995, 6.65956504988089730504825770078, 7.09027976863346583663906770029, 8.460558806753381159575475572854, 9.855559264285757365894589158273, 10.33649315983170256938586077098