L(s) = 1 | − 2-s − 3-s + 4-s − 1.63·5-s + 6-s − 1.98·7-s − 8-s + 9-s + 1.63·10-s − 11-s − 12-s + 3.77·13-s + 1.98·14-s + 1.63·15-s + 16-s − 4.54·17-s − 18-s + 3.03·19-s − 1.63·20-s + 1.98·21-s + 22-s + 0.766·23-s + 24-s − 2.31·25-s − 3.77·26-s − 27-s − 1.98·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.733·5-s + 0.408·6-s − 0.748·7-s − 0.353·8-s + 0.333·9-s + 0.518·10-s − 0.301·11-s − 0.288·12-s + 1.04·13-s + 0.529·14-s + 0.423·15-s + 0.250·16-s − 1.10·17-s − 0.235·18-s + 0.696·19-s − 0.366·20-s + 0.432·21-s + 0.213·22-s + 0.159·23-s + 0.204·24-s − 0.462·25-s − 0.741·26-s − 0.192·27-s − 0.374·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 + 1.63T + 5T^{2} \) |
| 7 | \( 1 + 1.98T + 7T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 4.54T + 17T^{2} \) |
| 19 | \( 1 - 3.03T + 19T^{2} \) |
| 23 | \( 1 - 0.766T + 23T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 + 1.58T + 37T^{2} \) |
| 41 | \( 1 + 6.07T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 + 5.55T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 7.18T + 59T^{2} \) |
| 67 | \( 1 - 8.89T + 67T^{2} \) |
| 71 | \( 1 + 9.98T + 71T^{2} \) |
| 73 | \( 1 - 9.46T + 73T^{2} \) |
| 79 | \( 1 - 5.54T + 79T^{2} \) |
| 83 | \( 1 - 2.25T + 83T^{2} \) |
| 89 | \( 1 + 7.45T + 89T^{2} \) |
| 97 | \( 1 + 1.35T + 97T^{2} \) |
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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
−8.164495189774067402575506838241, −7.30057772292096150465435820856, −6.71097194725388194143683127727, −6.05785849083861143454416865824, −5.20884681485770400554093345192, −4.13157366212424016756038735885, −3.44629403855785069758159091908, −2.38344072506593267356552540089, −1.05231468273781062762517470249, 0,
1.05231468273781062762517470249, 2.38344072506593267356552540089, 3.44629403855785069758159091908, 4.13157366212424016756038735885, 5.20884681485770400554093345192, 6.05785849083861143454416865824, 6.71097194725388194143683127727, 7.30057772292096150465435820856, 8.164495189774067402575506838241