| L(s) = 1 | + 3-s − 1.38·5-s − 0.871·7-s + 9-s − 5.74·11-s + 3.90·13-s − 1.38·15-s − 0.477·17-s + 2.16·19-s − 0.871·21-s + 3.53·23-s − 3.07·25-s + 27-s − 5.05·29-s + 3.41·31-s − 5.74·33-s + 1.21·35-s − 4.98·37-s + 3.90·39-s − 5.50·41-s − 3.83·43-s − 1.38·45-s − 13.6·47-s − 6.23·49-s − 0.477·51-s − 11.6·53-s + 7.98·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.621·5-s − 0.329·7-s + 0.333·9-s − 1.73·11-s + 1.08·13-s − 0.358·15-s − 0.115·17-s + 0.495·19-s − 0.190·21-s + 0.736·23-s − 0.614·25-s + 0.192·27-s − 0.938·29-s + 0.613·31-s − 1.00·33-s + 0.204·35-s − 0.820·37-s + 0.625·39-s − 0.860·41-s − 0.584·43-s − 0.207·45-s − 1.99·47-s − 0.891·49-s − 0.0669·51-s − 1.60·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 + 0.871T + 7T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 13 | \( 1 - 3.90T + 13T^{2} \) |
| 17 | \( 1 + 0.477T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 - 3.53T + 23T^{2} \) |
| 29 | \( 1 + 5.05T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 + 4.98T + 37T^{2} \) |
| 41 | \( 1 + 5.50T + 41T^{2} \) |
| 43 | \( 1 + 3.83T + 43T^{2} \) |
| 47 | \( 1 + 13.6T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 0.528T + 59T^{2} \) |
| 61 | \( 1 - 1.27T + 61T^{2} \) |
| 67 | \( 1 - 8.13T + 67T^{2} \) |
| 71 | \( 1 - 4.17T + 71T^{2} \) |
| 73 | \( 1 + 6.12T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 1.01T + 83T^{2} \) |
| 89 | \( 1 + 9.40T + 89T^{2} \) |
| 97 | \( 1 + 0.321T + 97T^{2} \) |
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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
−8.502093327911818181402676429314, −8.119185404356004154002279288876, −7.38026537193751277003205469189, −6.51190476727601604255819513059, −5.46428084943718152382330818906, −4.68048858821666511717350231746, −3.49467658768158127292921290469, −3.03358858914212224293941068619, −1.71105587019226403451880842145, 0,
1.71105587019226403451880842145, 3.03358858914212224293941068619, 3.49467658768158127292921290469, 4.68048858821666511717350231746, 5.46428084943718152382330818906, 6.51190476727601604255819513059, 7.38026537193751277003205469189, 8.119185404356004154002279288876, 8.502093327911818181402676429314
