| L(s) = 1 | + 1.39·2-s − 3-s − 0.0545·4-s − 2.50·5-s − 1.39·6-s + 4.61·7-s − 2.86·8-s + 9-s − 3.49·10-s + 6.41·11-s + 0.0545·12-s − 4.63·13-s + 6.44·14-s + 2.50·15-s − 3.88·16-s − 0.986·17-s + 1.39·18-s − 2.31·19-s + 0.136·20-s − 4.61·21-s + 8.94·22-s − 23-s + 2.86·24-s + 1.26·25-s − 6.45·26-s − 27-s − 0.252·28-s + ⋯ |
| L(s) = 1 | + 0.986·2-s − 0.577·3-s − 0.0272·4-s − 1.11·5-s − 0.569·6-s + 1.74·7-s − 1.01·8-s + 0.333·9-s − 1.10·10-s + 1.93·11-s + 0.0157·12-s − 1.28·13-s + 1.72·14-s + 0.646·15-s − 0.971·16-s − 0.239·17-s + 0.328·18-s − 0.532·19-s + 0.0305·20-s − 1.00·21-s + 1.90·22-s − 0.208·23-s + 0.584·24-s + 0.252·25-s − 1.26·26-s − 0.192·27-s − 0.0476·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.045973636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.045973636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 5 | \( 1 + 2.50T + 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 - 6.41T + 11T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 17 | \( 1 + 0.986T + 17T^{2} \) |
| 19 | \( 1 + 2.31T + 19T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 - 9.39T + 41T^{2} \) |
| 43 | \( 1 - 0.107T + 43T^{2} \) |
| 47 | \( 1 - 0.665T + 47T^{2} \) |
| 53 | \( 1 - 9.08T + 53T^{2} \) |
| 59 | \( 1 - 5.42T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 1.36T + 73T^{2} \) |
| 79 | \( 1 - 4.07T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 3.27T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−9.013999215719513593747627722080, −8.315509252938120448843744163310, −7.50564383823968595851711953502, −6.68767812312761461179107496875, −5.78431144483298844218670994547, −4.77798154843311481020158206243, −4.36209669843330326666678587891, −3.88227616226884025424920847323, −2.34368489516885002462140531545, −0.878616856861523771514271835241,
0.878616856861523771514271835241, 2.34368489516885002462140531545, 3.88227616226884025424920847323, 4.36209669843330326666678587891, 4.77798154843311481020158206243, 5.78431144483298844218670994547, 6.68767812312761461179107496875, 7.50564383823968595851711953502, 8.315509252938120448843744163310, 9.013999215719513593747627722080
