| L(s) = 1 | − 2-s + 1.72·3-s + 4-s + 0.546·5-s − 1.72·6-s + 2.37·7-s − 8-s − 0.0170·9-s − 0.546·10-s − 1.95·11-s + 1.72·12-s − 3.17·13-s − 2.37·14-s + 0.943·15-s + 16-s − 0.643·17-s + 0.0170·18-s + 0.483·19-s + 0.546·20-s + 4.11·21-s + 1.95·22-s − 23-s − 1.72·24-s − 4.70·25-s + 3.17·26-s − 5.21·27-s + 2.37·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.997·3-s + 0.5·4-s + 0.244·5-s − 0.705·6-s + 0.899·7-s − 0.353·8-s − 0.00567·9-s − 0.172·10-s − 0.589·11-s + 0.498·12-s − 0.880·13-s − 0.636·14-s + 0.243·15-s + 0.250·16-s − 0.155·17-s + 0.00401·18-s + 0.110·19-s + 0.122·20-s + 0.896·21-s + 0.416·22-s − 0.208·23-s − 0.352·24-s − 0.940·25-s + 0.622·26-s − 1.00·27-s + 0.449·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 - 0.546T + 5T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 + 1.95T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 + 0.643T + 17T^{2} \) |
| 19 | \( 1 - 0.483T + 19T^{2} \) |
| 29 | \( 1 - 6.04T + 29T^{2} \) |
| 31 | \( 1 + 4.23T + 31T^{2} \) |
| 37 | \( 1 + 6.91T + 37T^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 - 7.54T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 3.90T + 59T^{2} \) |
| 61 | \( 1 - 1.06T + 61T^{2} \) |
| 67 | \( 1 + 9.73T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 4.64T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 5.56T + 89T^{2} \) |
| 97 | \( 1 - 8.21T + 97T^{2} \) |
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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
−7.71241286200303897892594541856, −7.51727590103158553366244557867, −6.46753203607473443087879284831, −5.55029801011522306670932585382, −4.89439022192701498625067075507, −3.89183838912758223968561650937, −2.89734808145265020696444581968, −2.27882799826164816639175566307, −1.54940192013288765003172012897, 0,
1.54940192013288765003172012897, 2.27882799826164816639175566307, 2.89734808145265020696444581968, 3.89183838912758223968561650937, 4.89439022192701498625067075507, 5.55029801011522306670932585382, 6.46753203607473443087879284831, 7.51727590103158553366244557867, 7.71241286200303897892594541856
