| L(s) = 1 | + 2-s − 1.45·3-s + 4-s − 2.49·5-s − 1.45·6-s − 3.69·7-s + 8-s − 0.884·9-s − 2.49·10-s − 3.95·11-s − 1.45·12-s − 4.27·13-s − 3.69·14-s + 3.63·15-s + 16-s + 2.57·17-s − 0.884·18-s − 6.17·19-s − 2.49·20-s + 5.37·21-s − 3.95·22-s + 2.01·23-s − 1.45·24-s + 1.24·25-s − 4.27·26-s + 5.64·27-s − 3.69·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.839·3-s + 0.5·4-s − 1.11·5-s − 0.593·6-s − 1.39·7-s + 0.353·8-s − 0.294·9-s − 0.790·10-s − 1.19·11-s − 0.419·12-s − 1.18·13-s − 0.988·14-s + 0.938·15-s + 0.250·16-s + 0.624·17-s − 0.208·18-s − 1.41·19-s − 0.558·20-s + 1.17·21-s − 0.844·22-s + 0.420·23-s − 0.296·24-s + 0.249·25-s − 0.837·26-s + 1.08·27-s − 0.698·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1413265244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1413265244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 1.45T + 3T^{2} \) |
| 5 | \( 1 + 2.49T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + 3.95T + 11T^{2} \) |
| 13 | \( 1 + 4.27T + 13T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 + 6.17T + 19T^{2} \) |
| 23 | \( 1 - 2.01T + 23T^{2} \) |
| 29 | \( 1 - 0.563T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 + 5.61T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 - 3.56T + 47T^{2} \) |
| 53 | \( 1 + 14.2T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 9.71T + 61T^{2} \) |
| 67 | \( 1 + 0.780T + 67T^{2} \) |
| 71 | \( 1 - 7.96T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 5.77T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 8.78T + 97T^{2} \) |
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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
−8.201685224531966938957317436171, −7.56537505649977891107080993834, −6.76139728270041226735958388410, −6.29186659252134631773349896010, −5.20303352925939932726210998555, −4.98321280865586030846474422192, −3.74445605803843875199793093374, −3.20882923329181427714226792018, −2.27949503355743229446097614713, −0.18337251635239796307508901994,
0.18337251635239796307508901994, 2.27949503355743229446097614713, 3.20882923329181427714226792018, 3.74445605803843875199793093374, 4.98321280865586030846474422192, 5.20303352925939932726210998555, 6.29186659252134631773349896010, 6.76139728270041226735958388410, 7.56537505649977891107080993834, 8.201685224531966938957317436171
