| L(s) = 1 | − 2-s − 1.88·3-s + 4-s − 5-s + 1.88·6-s + 1.11·7-s − 8-s + 0.563·9-s + 10-s − 6.32·11-s − 1.88·12-s − 2.82·13-s − 1.11·14-s + 1.88·15-s + 16-s + 3.93·17-s − 0.563·18-s − 1.08·19-s − 20-s − 2.11·21-s + 6.32·22-s + 5.37·23-s + 1.88·24-s + 25-s + 2.82·26-s + 4.59·27-s + 1.11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.08·3-s + 0.5·4-s − 0.447·5-s + 0.770·6-s + 0.422·7-s − 0.353·8-s + 0.187·9-s + 0.316·10-s − 1.90·11-s − 0.544·12-s − 0.783·13-s − 0.298·14-s + 0.487·15-s + 0.250·16-s + 0.953·17-s − 0.132·18-s − 0.248·19-s − 0.223·20-s − 0.460·21-s + 1.34·22-s + 1.12·23-s + 0.385·24-s + 0.200·25-s + 0.554·26-s + 0.885·27-s + 0.211·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 1.88T + 3T^{2} \) |
| 7 | \( 1 - 1.11T + 7T^{2} \) |
| 11 | \( 1 + 6.32T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 + 1.08T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 - 3.07T + 29T^{2} \) |
| 31 | \( 1 - 3.88T + 31T^{2} \) |
| 37 | \( 1 + 0.283T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 6.23T + 47T^{2} \) |
| 53 | \( 1 + 0.293T + 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 4.56T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 6.28T + 83T^{2} \) |
| 89 | \( 1 - 7.99T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−8.158256034550464702356660016279, −7.39026212021026847582299263128, −6.77542373434459617315707499581, −5.75995826338345456882641106336, −5.16073323250555564945427384268, −4.65359010862600743843223374229, −3.14438477323698152828718673937, −2.44228088787066898959849467176, −0.971848507756291376257836399583, 0,
0.971848507756291376257836399583, 2.44228088787066898959849467176, 3.14438477323698152828718673937, 4.65359010862600743843223374229, 5.16073323250555564945427384268, 5.75995826338345456882641106336, 6.77542373434459617315707499581, 7.39026212021026847582299263128, 8.158256034550464702356660016279
