L(s) = 1 | − 2-s + 0.888·3-s + 4-s + 2.86·5-s − 0.888·6-s − 2.14·7-s − 8-s − 2.21·9-s − 2.86·10-s + 1.49·11-s + 0.888·12-s + 6.87·13-s + 2.14·14-s + 2.54·15-s + 16-s − 7.83·17-s + 2.21·18-s − 1.29·19-s + 2.86·20-s − 1.90·21-s − 1.49·22-s + 23-s − 0.888·24-s + 3.23·25-s − 6.87·26-s − 4.62·27-s − 2.14·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.512·3-s + 0.5·4-s + 1.28·5-s − 0.362·6-s − 0.812·7-s − 0.353·8-s − 0.736·9-s − 0.907·10-s + 0.450·11-s + 0.256·12-s + 1.90·13-s + 0.574·14-s + 0.658·15-s + 0.250·16-s − 1.89·17-s + 0.521·18-s − 0.298·19-s + 0.641·20-s − 0.416·21-s − 0.318·22-s + 0.208·23-s − 0.181·24-s + 0.646·25-s − 1.34·26-s − 0.890·27-s − 0.406·28-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 | \( 1 - 0.888T + 3T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 - 1.49T + 11T^{2} \) |
| 13 | \( 1 - 6.87T + 13T^{2} \) |
| 17 | \( 1 + 7.83T + 17T^{2} \) |
| 19 | \( 1 + 1.29T + 19T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 4.87T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 8.21T + 47T^{2} \) |
| 53 | \( 1 - 4.48T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 - 6.94T + 71T^{2} \) |
| 73 | \( 1 - 6.40T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 3.93T + 83T^{2} \) |
| 89 | \( 1 - 5.77T + 89T^{2} \) |
| 97 | \( 1 - 1.86T + 97T^{2} \) |
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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
−8.048678892409513157020003131609, −6.64564854360261166010457417306, −6.50880057906194339314710018857, −5.95156265949089530264179334621, −4.96018278128459055844132902617, −3.69520872995166448800800166822, −3.12205594843432768536118306853, −2.11784959030101932539113389304, −1.53153741202928632495112382485, 0,
1.53153741202928632495112382485, 2.11784959030101932539113389304, 3.12205594843432768536118306853, 3.69520872995166448800800166822, 4.96018278128459055844132902617, 5.95156265949089530264179334621, 6.50880057906194339314710018857, 6.64564854360261166010457417306, 8.048678892409513157020003131609