| L(s) = 1 | − 0.697·3-s + 5-s − 4.99·7-s − 2.51·9-s + 4.56·11-s − 0.548·13-s − 0.697·15-s − 4.13·17-s + 4.54·19-s + 3.48·21-s − 3.56·23-s + 25-s + 3.84·27-s + 4.58·29-s − 3.18·31-s − 3.18·33-s − 4.99·35-s + 11.2·37-s + 0.382·39-s + 1.69·41-s + 7.02·43-s − 2.51·45-s − 7.52·47-s + 17.9·49-s + 2.88·51-s − 10.0·53-s + 4.56·55-s + ⋯ |
| L(s) = 1 | − 0.402·3-s + 0.447·5-s − 1.88·7-s − 0.837·9-s + 1.37·11-s − 0.151·13-s − 0.180·15-s − 1.00·17-s + 1.04·19-s + 0.759·21-s − 0.744·23-s + 0.200·25-s + 0.740·27-s + 0.851·29-s − 0.572·31-s − 0.554·33-s − 0.843·35-s + 1.84·37-s + 0.0612·39-s + 0.264·41-s + 1.07·43-s − 0.374·45-s − 1.09·47-s + 2.55·49-s + 0.403·51-s − 1.38·53-s + 0.615·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 0.697T + 3T^{2} \) |
| 7 | \( 1 + 4.99T + 7T^{2} \) |
| 11 | \( 1 - 4.56T + 11T^{2} \) |
| 13 | \( 1 + 0.548T + 13T^{2} \) |
| 17 | \( 1 + 4.13T + 17T^{2} \) |
| 19 | \( 1 - 4.54T + 19T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 + 3.18T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 1.69T + 41T^{2} \) |
| 43 | \( 1 - 7.02T + 43T^{2} \) |
| 47 | \( 1 + 7.52T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 2.20T + 59T^{2} \) |
| 61 | \( 1 + 7.93T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 6.00T + 83T^{2} \) |
| 89 | \( 1 - 1.91T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.25098994336543408461017937311, −6.49742359181573060995488979043, −6.24780051091963194016873705665, −5.69305793857823266405801301040, −4.64875677326339098990212637697, −3.82177634153091308759667988742, −3.07876420761009099526204066109, −2.41306929295503765517590264021, −1.06242780413744380301265070950, 0,
1.06242780413744380301265070950, 2.41306929295503765517590264021, 3.07876420761009099526204066109, 3.82177634153091308759667988742, 4.64875677326339098990212637697, 5.69305793857823266405801301040, 6.24780051091963194016873705665, 6.49742359181573060995488979043, 7.25098994336543408461017937311
