L(s) = 1 | + 2-s − 1.80·3-s + 4-s − 2.44·5-s − 1.80·6-s − 4.49·7-s + 8-s + 0.246·9-s − 2.44·10-s − 2.49·11-s − 1.80·12-s − 4.49·14-s + 4.40·15-s + 16-s + 7.74·17-s + 0.246·18-s + 19-s − 2.44·20-s + 8.09·21-s − 2.49·22-s + 4.27·23-s − 1.80·24-s + 0.978·25-s + 4.96·27-s − 4.49·28-s − 0.713·29-s + 4.40·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.04·3-s + 0.5·4-s − 1.09·5-s − 0.735·6-s − 1.69·7-s + 0.353·8-s + 0.0823·9-s − 0.773·10-s − 0.751·11-s − 0.520·12-s − 1.20·14-s + 1.13·15-s + 0.250·16-s + 1.87·17-s + 0.0582·18-s + 0.229·19-s − 0.546·20-s + 1.76·21-s − 0.531·22-s + 0.891·23-s − 0.367·24-s + 0.195·25-s + 0.954·27-s − 0.849·28-s − 0.132·29-s + 0.804·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 1.80T + 3T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 + 4.49T + 7T^{2} \) |
| 11 | \( 1 + 2.49T + 11T^{2} \) |
| 17 | \( 1 - 7.74T + 17T^{2} \) |
| 23 | \( 1 - 4.27T + 23T^{2} \) |
| 29 | \( 1 + 0.713T + 29T^{2} \) |
| 31 | \( 1 + 0.911T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 + 3.28T + 41T^{2} \) |
| 43 | \( 1 - 7.20T + 43T^{2} \) |
| 47 | \( 1 - 5.20T + 47T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 3.43T + 61T^{2} \) |
| 67 | \( 1 + 1.40T + 67T^{2} \) |
| 71 | \( 1 - 3.86T + 71T^{2} \) |
| 73 | \( 1 + 7.74T + 73T^{2} \) |
| 79 | \( 1 + 1.91T + 79T^{2} \) |
| 83 | \( 1 + 0.396T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 3.16T + 97T^{2} \) |
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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
−7.48426960782067434239553571275, −6.86954603158288145956183257735, −6.00048796917132412788004119609, −5.68732318380046078431183034393, −4.88283699739472147420776504736, −4.00046129299320099414913390715, −3.19476299705269266927594824794, −2.84575033684687096113167131324, −0.955247731925692785920651606857, 0,
0.955247731925692785920651606857, 2.84575033684687096113167131324, 3.19476299705269266927594824794, 4.00046129299320099414913390715, 4.88283699739472147420776504736, 5.68732318380046078431183034393, 6.00048796917132412788004119609, 6.86954603158288145956183257735, 7.48426960782067434239553571275