L(s) = 1 | − 2-s − 3-s + 4-s − 2.61·5-s + 6-s − 0.977·7-s − 8-s + 9-s + 2.61·10-s − 0.240·11-s − 12-s + 13-s + 0.977·14-s + 2.61·15-s + 16-s − 4.91·17-s − 18-s − 1.97·19-s − 2.61·20-s + 0.977·21-s + 0.240·22-s − 0.244·23-s + 24-s + 1.84·25-s − 26-s − 27-s − 0.977·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.17·5-s + 0.408·6-s − 0.369·7-s − 0.353·8-s + 0.333·9-s + 0.827·10-s − 0.0725·11-s − 0.288·12-s + 0.277·13-s + 0.261·14-s + 0.675·15-s + 0.250·16-s − 1.19·17-s − 0.235·18-s − 0.453·19-s − 0.585·20-s + 0.213·21-s + 0.0513·22-s − 0.0510·23-s + 0.204·24-s + 0.369·25-s − 0.196·26-s − 0.192·27-s − 0.184·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 2.61T + 5T^{2} \) |
| 7 | \( 1 + 0.977T + 7T^{2} \) |
| 11 | \( 1 + 0.240T + 11T^{2} \) |
| 17 | \( 1 + 4.91T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 23 | \( 1 + 0.244T + 23T^{2} \) |
| 29 | \( 1 - 8.09T + 29T^{2} \) |
| 31 | \( 1 + 7.15T + 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 5.03T + 43T^{2} \) |
| 47 | \( 1 - 9.26T + 47T^{2} \) |
| 53 | \( 1 - 9.79T + 53T^{2} \) |
| 59 | \( 1 + 7.05T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 9.42T + 71T^{2} \) |
| 73 | \( 1 - 0.976T + 73T^{2} \) |
| 79 | \( 1 + 8.87T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 0.490T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.56840468531176288075754375194, −6.83212103589310422302483872374, −6.33399012445370351077253595210, −5.53969200797583123293927968179, −4.46160173985057713102616376876, −4.05073596685526856900119702179, −3.05069653403504079397863069218, −2.13215641186555478645179106778, −0.866625426477179672904931116611, 0,
0.866625426477179672904931116611, 2.13215641186555478645179106778, 3.05069653403504079397863069218, 4.05073596685526856900119702179, 4.46160173985057713102616376876, 5.53969200797583123293927968179, 6.33399012445370351077253595210, 6.83212103589310422302483872374, 7.56840468531176288075754375194