L(s) = 1 | − 2-s + 3-s + 4-s + 2.41·5-s − 6-s − 7-s − 8-s + 9-s − 2.41·10-s − 3.24·11-s + 12-s + 14-s + 2.41·15-s + 16-s + 1.80·17-s − 18-s + 2.73·19-s + 2.41·20-s − 21-s + 3.24·22-s − 6.58·23-s − 24-s + 0.832·25-s + 27-s − 28-s + 6.22·29-s − 2.41·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.08·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.763·10-s − 0.979·11-s + 0.288·12-s + 0.267·14-s + 0.623·15-s + 0.250·16-s + 0.437·17-s − 0.235·18-s + 0.628·19-s + 0.540·20-s − 0.218·21-s + 0.692·22-s − 1.37·23-s − 0.204·24-s + 0.166·25-s + 0.192·27-s − 0.188·28-s + 1.15·29-s − 0.440·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2.41T + 5T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 17 | \( 1 - 1.80T + 17T^{2} \) |
| 19 | \( 1 - 2.73T + 19T^{2} \) |
| 23 | \( 1 + 6.58T + 23T^{2} \) |
| 29 | \( 1 - 6.22T + 29T^{2} \) |
| 31 | \( 1 + 2.72T + 31T^{2} \) |
| 37 | \( 1 + 8.93T + 37T^{2} \) |
| 41 | \( 1 + 9.15T + 41T^{2} \) |
| 43 | \( 1 + 3.95T + 43T^{2} \) |
| 47 | \( 1 + 9.77T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 - 2.54T + 67T^{2} \) |
| 71 | \( 1 - 1.50T + 71T^{2} \) |
| 73 | \( 1 - 0.786T + 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 - 3.51T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.81935057388966980892118805238, −6.90356370545512699588145001458, −6.36583173762212813165483558650, −5.51042579217296769371572265264, −4.97041766145990188400380928316, −3.63884544792054286722307415143, −2.97955454978843168009570220574, −2.13193394247118709772878554531, −1.48407580225627986325243838214, 0,
1.48407580225627986325243838214, 2.13193394247118709772878554531, 2.97955454978843168009570220574, 3.63884544792054286722307415143, 4.97041766145990188400380928316, 5.51042579217296769371572265264, 6.36583173762212813165483558650, 6.90356370545512699588145001458, 7.81935057388966980892118805238