| L(s) = 1 | + 0.378·3-s − 4.27·5-s − 4.43·7-s − 2.85·9-s − 2.38·11-s + 13-s − 1.62·15-s − 3.51·17-s + 7.74·19-s − 1.67·21-s − 1.35·23-s + 13.2·25-s − 2.21·27-s + 0.643·29-s − 7.63·31-s − 0.905·33-s + 18.9·35-s + 5.09·37-s + 0.378·39-s − 5.19·41-s + 1.19·43-s + 12.2·45-s + 2.09·47-s + 12.6·49-s − 1.33·51-s + 9.19·53-s + 10.2·55-s + ⋯ |
| L(s) = 1 | + 0.218·3-s − 1.91·5-s − 1.67·7-s − 0.952·9-s − 0.720·11-s + 0.277·13-s − 0.418·15-s − 0.853·17-s + 1.77·19-s − 0.366·21-s − 0.282·23-s + 2.65·25-s − 0.427·27-s + 0.119·29-s − 1.37·31-s − 0.157·33-s + 3.20·35-s + 0.838·37-s + 0.0606·39-s − 0.811·41-s + 0.182·43-s + 1.82·45-s + 0.305·47-s + 1.80·49-s − 0.186·51-s + 1.26·53-s + 1.37·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4380829314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4380829314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.378T + 3T^{2} \) |
| 5 | \( 1 + 4.27T + 5T^{2} \) |
| 7 | \( 1 + 4.43T + 7T^{2} \) |
| 11 | \( 1 + 2.38T + 11T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 - 7.74T + 19T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 - 0.643T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 - 9.19T + 53T^{2} \) |
| 59 | \( 1 + 2.38T + 59T^{2} \) |
| 61 | \( 1 + 3.68T + 61T^{2} \) |
| 67 | \( 1 + 9.36T + 67T^{2} \) |
| 71 | \( 1 + 9.96T + 71T^{2} \) |
| 73 | \( 1 - 6.26T + 73T^{2} \) |
| 79 | \( 1 - 0.777T + 79T^{2} \) |
| 83 | \( 1 + 7.76T + 83T^{2} \) |
| 89 | \( 1 + 7.61T + 89T^{2} \) |
| 97 | \( 1 - 8.97T + 97T^{2} \) |
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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
−9.133482450373126231868926688775, −8.643065681912566251684364337444, −7.62979640908105475922766698345, −7.24578993007610007048365975007, −6.21085064742135562423680126910, −5.25988545531743718881019667629, −4.06292503136146786017990641253, −3.31940245358544435839949988333, −2.81919016637088894864324213115, −0.42386886155492492061098730422,
0.42386886155492492061098730422, 2.81919016637088894864324213115, 3.31940245358544435839949988333, 4.06292503136146786017990641253, 5.25988545531743718881019667629, 6.21085064742135562423680126910, 7.24578993007610007048365975007, 7.62979640908105475922766698345, 8.643065681912566251684364337444, 9.133482450373126231868926688775
