| L(s) = 1 | − 0.0520·2-s + 1.79·3-s − 1.99·4-s + 5-s − 0.0932·6-s − 4.01·7-s + 0.208·8-s + 0.210·9-s − 0.0520·10-s − 3.57·12-s + 13-s + 0.208·14-s + 1.79·15-s + 3.98·16-s − 3.51·17-s − 0.0109·18-s + 5.87·19-s − 1.99·20-s − 7.18·21-s + 5.34·23-s + 0.372·24-s + 25-s − 0.0520·26-s − 4.99·27-s + 8.01·28-s − 6.81·29-s − 0.0932·30-s + ⋯ |
| L(s) = 1 | − 0.0368·2-s + 1.03·3-s − 0.998·4-s + 0.447·5-s − 0.0380·6-s − 1.51·7-s + 0.0735·8-s + 0.0702·9-s − 0.0164·10-s − 1.03·12-s + 0.277·13-s + 0.0557·14-s + 0.462·15-s + 0.995·16-s − 0.851·17-s − 0.00258·18-s + 1.34·19-s − 0.446·20-s − 1.56·21-s + 1.11·23-s + 0.0760·24-s + 0.200·25-s − 0.0102·26-s − 0.961·27-s + 1.51·28-s − 1.26·29-s − 0.0170·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0520T + 2T^{2} \) |
| 3 | \( 1 - 1.79T + 3T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 - 5.34T + 23T^{2} \) |
| 29 | \( 1 + 6.81T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 + 8.66T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 1.89T + 53T^{2} \) |
| 59 | \( 1 - 6.49T + 59T^{2} \) |
| 61 | \( 1 + 0.835T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 6.25T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 9.00T + 83T^{2} \) |
| 89 | \( 1 + 8.81T + 89T^{2} \) |
| 97 | \( 1 + 7.86T + 97T^{2} \) |
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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
−7.42842258843365682788648451200, −7.04673711646059573043850210591, −5.93362060767146192103672368106, −5.56658536016032197750491178327, −4.51683795277036317228854010929, −3.70857378819280335949604556322, −3.13185307304030554052602340793, −2.55006786093548856930743619257, −1.21740348999739204285027502562, 0,
1.21740348999739204285027502562, 2.55006786093548856930743619257, 3.13185307304030554052602340793, 3.70857378819280335949604556322, 4.51683795277036317228854010929, 5.56658536016032197750491178327, 5.93362060767146192103672368106, 7.04673711646059573043850210591, 7.42842258843365682788648451200
