| L(s) = 1 | − 1.74·3-s + 0.394·5-s − 1.78·7-s + 0.0406·9-s − 3.20·11-s − 5.98·13-s − 0.688·15-s − 17-s − 3.98·19-s + 3.10·21-s − 1.34·23-s − 4.84·25-s + 5.16·27-s + 2.92·29-s − 2.76·31-s + 5.59·33-s − 0.703·35-s − 7.06·37-s + 10.4·39-s − 3.64·41-s + 10.2·43-s + 0.0160·45-s − 2.68·47-s − 3.82·49-s + 1.74·51-s − 12.7·53-s − 1.26·55-s + ⋯ |
| L(s) = 1 | − 1.00·3-s + 0.176·5-s − 0.673·7-s + 0.0135·9-s − 0.967·11-s − 1.66·13-s − 0.177·15-s − 0.242·17-s − 0.913·19-s + 0.678·21-s − 0.279·23-s − 0.968·25-s + 0.993·27-s + 0.543·29-s − 0.496·31-s + 0.973·33-s − 0.118·35-s − 1.16·37-s + 1.67·39-s − 0.569·41-s + 1.56·43-s + 0.00239·45-s − 0.391·47-s − 0.546·49-s + 0.244·51-s − 1.74·53-s − 0.170·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2740930589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2740930589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 1.74T + 3T^{2} \) |
| 5 | \( 1 - 0.394T + 5T^{2} \) |
| 7 | \( 1 + 1.78T + 7T^{2} \) |
| 11 | \( 1 + 3.20T + 11T^{2} \) |
| 13 | \( 1 + 5.98T + 13T^{2} \) |
| 19 | \( 1 + 3.98T + 19T^{2} \) |
| 23 | \( 1 + 1.34T + 23T^{2} \) |
| 29 | \( 1 - 2.92T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 + 7.06T + 37T^{2} \) |
| 41 | \( 1 + 3.64T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 61 | \( 1 - 7.03T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 2.56T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 9.29T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 7.04T + 97T^{2} \) |
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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
−8.311345007229716322139371500003, −7.69088391839894733932204657706, −6.72202287872370690140808031519, −6.33367421530760196262949835623, −5.27933015375784902223987854925, −5.06608059640730232159223077962, −3.95677313011319846864642217588, −2.80802529101590643307444614257, −2.07898260879400512260308246091, −0.29117593311715403109197463608,
0.29117593311715403109197463608, 2.07898260879400512260308246091, 2.80802529101590643307444614257, 3.95677313011319846864642217588, 5.06608059640730232159223077962, 5.27933015375784902223987854925, 6.33367421530760196262949835623, 6.72202287872370690140808031519, 7.69088391839894733932204657706, 8.311345007229716322139371500003
