| L(s) = 1 | − 1.24·2-s − 2.42·3-s − 0.443·4-s + 0.848·5-s + 3.02·6-s − 1.14·7-s + 3.04·8-s + 2.87·9-s − 1.05·10-s + 6.08·11-s + 1.07·12-s − 13-s + 1.42·14-s − 2.05·15-s − 2.91·16-s − 7.37·17-s − 3.58·18-s + 0.506·19-s − 0.376·20-s + 2.76·21-s − 7.58·22-s − 1.14·23-s − 7.39·24-s − 4.27·25-s + 1.24·26-s + 0.297·27-s + 0.505·28-s + ⋯ |
| L(s) = 1 | − 0.882·2-s − 1.39·3-s − 0.221·4-s + 0.379·5-s + 1.23·6-s − 0.430·7-s + 1.07·8-s + 0.959·9-s − 0.334·10-s + 1.83·11-s + 0.310·12-s − 0.277·13-s + 0.380·14-s − 0.531·15-s − 0.729·16-s − 1.78·17-s − 0.846·18-s + 0.116·19-s − 0.0841·20-s + 0.603·21-s − 1.61·22-s − 0.238·23-s − 1.50·24-s − 0.855·25-s + 0.244·26-s + 0.0571·27-s + 0.0955·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 3 | \( 1 + 2.42T + 3T^{2} \) |
| 5 | \( 1 - 0.848T + 5T^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 - 6.08T + 11T^{2} \) |
| 17 | \( 1 + 7.37T + 17T^{2} \) |
| 19 | \( 1 - 0.506T + 19T^{2} \) |
| 23 | \( 1 + 1.14T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 37 | \( 1 - 7.75T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 + 9.58T + 43T^{2} \) |
| 47 | \( 1 + 5.58T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 3.26T + 61T^{2} \) |
| 67 | \( 1 - 7.13T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 9.30T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−10.92800748668269195733727153730, −9.561878054314148948120131381784, −9.467796039954213420175210268229, −8.170053487364666512150136663122, −6.71030626004297821641032361944, −6.35969396511805205627433034591, −4.98965437039789436900893820291, −4.04020528279066032307437325403, −1.60986859223855936996153615685, 0,
1.60986859223855936996153615685, 4.04020528279066032307437325403, 4.98965437039789436900893820291, 6.35969396511805205627433034591, 6.71030626004297821641032361944, 8.170053487364666512150136663122, 9.467796039954213420175210268229, 9.561878054314148948120131381784, 10.92800748668269195733727153730
