| L(s) = 1 | + 2.02·2-s + 1.05·3-s + 2.08·4-s − 5-s + 2.12·6-s − 1.12·7-s + 0.177·8-s − 1.89·9-s − 2.02·10-s − 2.80·11-s + 2.19·12-s − 6.07·13-s − 2.26·14-s − 1.05·15-s − 3.81·16-s + 1.23·17-s − 3.83·18-s − 5.53·19-s − 2.08·20-s − 1.17·21-s − 5.66·22-s + 9.28·23-s + 0.186·24-s + 25-s − 12.2·26-s − 5.14·27-s − 2.34·28-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 0.606·3-s + 1.04·4-s − 0.447·5-s + 0.867·6-s − 0.424·7-s + 0.0628·8-s − 0.631·9-s − 0.639·10-s − 0.844·11-s + 0.633·12-s − 1.68·13-s − 0.606·14-s − 0.271·15-s − 0.954·16-s + 0.299·17-s − 0.903·18-s − 1.26·19-s − 0.466·20-s − 0.257·21-s − 1.20·22-s + 1.93·23-s + 0.0381·24-s + 0.200·25-s − 2.40·26-s − 0.990·27-s − 0.442·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 3 | \( 1 - 1.05T + 3T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 11 | \( 1 + 2.80T + 11T^{2} \) |
| 13 | \( 1 + 6.07T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 23 | \( 1 - 9.28T + 23T^{2} \) |
| 29 | \( 1 - 4.03T + 29T^{2} \) |
| 31 | \( 1 + 0.0965T + 31T^{2} \) |
| 37 | \( 1 + 6.99T + 37T^{2} \) |
| 41 | \( 1 - 5.83T + 41T^{2} \) |
| 43 | \( 1 - 4.51T + 43T^{2} \) |
| 47 | \( 1 - 2.14T + 47T^{2} \) |
| 53 | \( 1 + 9.19T + 53T^{2} \) |
| 59 | \( 1 - 5.94T + 59T^{2} \) |
| 61 | \( 1 + 0.194T + 61T^{2} \) |
| 67 | \( 1 + 3.38T + 67T^{2} \) |
| 71 | \( 1 - 7.89T + 71T^{2} \) |
| 73 | \( 1 - 5.22T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 0.0878T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 0.266T + 97T^{2} \) |
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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
−8.761071337914656739610118055058, −7.83653459479210339553620391059, −7.07706404076496224159331083558, −6.27377717599277458002790267842, −5.19500164656953734896741359224, −4.80818112849109233656179717766, −3.73106702939775157608186142165, −2.84281848032047143597871705675, −2.46649654683765628609067714899, 0,
2.46649654683765628609067714899, 2.84281848032047143597871705675, 3.73106702939775157608186142165, 4.80818112849109233656179717766, 5.19500164656953734896741359224, 6.27377717599277458002790267842, 7.07706404076496224159331083558, 7.83653459479210339553620391059, 8.761071337914656739610118055058
