L(s) = 1 | − 2-s − 1.41·3-s + 4-s + 5-s + 1.41·6-s − 2.82·7-s − 8-s − 0.999·9-s − 10-s − 2·11-s − 1.41·12-s − 0.585·13-s + 2.82·14-s − 1.41·15-s + 16-s + 1.41·17-s + 0.999·18-s − 2·19-s + 20-s + 4.00·21-s + 2·22-s + 3.41·23-s + 1.41·24-s + 25-s + 0.585·26-s + 5.65·27-s − 2.82·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.816·3-s + 0.5·4-s + 0.447·5-s + 0.577·6-s − 1.06·7-s − 0.353·8-s − 0.333·9-s − 0.316·10-s − 0.603·11-s − 0.408·12-s − 0.162·13-s + 0.755·14-s − 0.365·15-s + 0.250·16-s + 0.342·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.872·21-s + 0.426·22-s + 0.711·23-s + 0.288·24-s + 0.200·25-s + 0.114·26-s + 1.08·27-s − 0.534·28-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 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3.41T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 1.17T + 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 - 0.343T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 3.07T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.201342374485404855248049227451, −7.21761558155977518386518425241, −6.58906575102368516064154302105, −5.95452393832042527191465372701, −5.39053380678540268544719861441, −4.38904211415536314128953945640, −3.05095241014274186966187772828, −2.54593829013625724921113772202, −1.05444077890884971116438749647, 0,
1.05444077890884971116438749647, 2.54593829013625724921113772202, 3.05095241014274186966187772828, 4.38904211415536314128953945640, 5.39053380678540268544719861441, 5.95452393832042527191465372701, 6.58906575102368516064154302105, 7.21761558155977518386518425241, 8.201342374485404855248049227451